Conservation and Innovation : Helmholtz's Struggle with Energy ...
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Fabio Bevilacqua<br />
<strong>Conservation</strong> <strong>and</strong> <strong>Innovation</strong> : <strong>Helmholtz's</strong> <strong>Struggle</strong> <strong>with</strong><br />
<strong>Energy</strong> Problems (1845-1894) <strong>and</strong> the Birth of Theoretical<br />
Physics<br />
Introduction .......................................................................................... 1<br />
From Physiology to Physics (1845-46).................................................. 11<br />
The Erhaltung <strong>and</strong> its two different conceptual roots: the<br />
impossibility of perpetual motion <strong>and</strong> the Newtonian forces<br />
hypothesis.(1847) ................................................................................. 22<br />
1 The "Einleitung": intelligibility of nature <strong>and</strong> conceptual<br />
explanation ................................................................................. 24<br />
2 The two roots of the vis viva principle <strong>and</strong> their supposed<br />
equivalence................................................................................. 30<br />
3 The duck, the rabbit <strong>and</strong> the principle of energy<br />
conservation................................................................................ 38<br />
4 An easy start: Mechanics ......................................................... 46<br />
5 Force equivalent of heat: a theoretical approach ....................... 47<br />
6 Electricity, galvanism <strong>and</strong> thermo-electric currents:<br />
Helmholtz <strong>and</strong> the batteries......................................................... 56<br />
7 Which force equivalents for magnetism <strong>and</strong><br />
electromagnetism?....................................................................... 62<br />
8 Conclusion............................................................................... 66<br />
Questioning one root: the central force controversy <strong>with</strong> Clausius<br />
(1852-54).............................................................................................. 68<br />
Popular conservation of "force" : where have the central forces<br />
gone ? (1854-64)................................................................................... 83<br />
The first history of the conservation principle: "evolution <strong>and</strong><br />
development, not discovery" (1862-65)................................................. 100<br />
A new tool : potential theory (1852-72) ................................................ 103<br />
An energy battlefield for the electrodynamic debate (1870-75).............. 104<br />
Unification, again ( 1884-94) ................................................................ 114<br />
What had been achieved: a turn of the century viewpoint ...................... 117<br />
Contemporary Historiography............................................................... 126<br />
Bibliography ......................................................................................... 144
Introduction 1<br />
Attempts at finding something in nature that is conserved through changes<br />
are very old. At the beginnings of the Greek speculations they were connected<br />
<strong>with</strong> discussions on the being <strong>and</strong> becoming, later <strong>with</strong> the ones about the<br />
impossibility of perpetual motion, in more recent times <strong>with</strong> the role of the<br />
principle of causality, the conservation of vis viva <strong>and</strong> of momentum, the<br />
definition of the concept of work. During the 1840s <strong>and</strong> early 1850s numerous<br />
formulations appeared of the "conversion" among natural phenomena <strong>and</strong> the<br />
"conservation" of something underlying them. By the 1860, despite a certain<br />
persistence of the term "force" ("Kraft") among a few German physicists, the<br />
term energy was generally adopted, although it did not assume an unequivocal<br />
meaning. 2. Histories rapidly written by the main actors 3, controversies about the<br />
1 Financial support for this research has been provided by the Consiglio<br />
Nazionale delle Ricerche. Earlier version of this essay were presented in Urbino<br />
(1989), Munich (1990), Chicago (1990). I wish to thank David Cahan, Rod<br />
Home, Stefan Wolff, Ivor Grattan Guinness, Enrico Giannetto <strong>and</strong> my wife<br />
Leitha for their comments on earlier versions of the manuscript. A shorter version<br />
of this essay, centered on the 1847 Erhaltung, is forthcoming in D.Cahan (ed):<br />
Helmholtz Scientist <strong>and</strong> Philosopher, California U.P.<br />
2Helmholtz recognised the equivalence between his own principle of conservation of<br />
"force" (1847) <strong>and</strong> Rankine's conservation of "energy" (1853) as early as 1856 in the "Bericht<br />
aus dem Jahre 1853". In Fort d Ph 9 (1856): 404-32, even if in relation to the idea of<br />
correlation he kept using the term "Kraft".<br />
3 See for instance the series of <strong>Helmholtz's</strong> "Bericht" on the theory of heat:<br />
Helmholtz, Hermann. "Bericht über “die Theorie der Wärme“ betreffende Arbeiten aus dem<br />
Jahre 1850-1851". In Fort d Ph im Jahre 1850-1851 7 (1855): 561-98; "Bericht aus dem<br />
Jahre 1852". In Fort d Ph im Jahre 1852 8 (1855): 369-87; "Bericht aus dem Jahre 1853". In<br />
Fort d Ph 9 (1856): 404-32; "Bericht aus dem Jahre 1854". In Fort d Ph 10 (1857): 361-98;<br />
"Bericht aus dem Jahre 1855". In Fort d Ph 11 (1858): 361-73; "Bericht aus dem Jahre<br />
1856". In Fort d Ph 12 (1859): 343-59.
different formulations 4 <strong>and</strong> priority debates 5 immediately followed. The term<br />
"energy" did not have, in fact, a univocal interpretation. Other debates,<br />
concentrated in the last two decades of the century, showed that these ideas had<br />
grown, spread to all branches of physics <strong>and</strong> raised competing research<br />
programs 6. During this period first-rank physicists wrote on the history <strong>and</strong><br />
meaning of the conservation ideas books that will here be defined as "classics" 7.<br />
4 Like the one between Clausius <strong>and</strong> Helmholtz in the Annalen (1852-54): Clausius,<br />
Rudolf. "On the Mechanical Equivalent of an Electric Discharge, <strong>and</strong> the Heating of the<br />
Conducting Wire which accompanies it." In Tyndall <strong>and</strong> Francis Scientific Memoirs on<br />
Natural Philosophy 1 (1853): 1-32 <strong>and</strong> 200-9; Helmholtz, Hermann. "Ueber einige Gesetze<br />
der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-<br />
elektrischen Versuche." In Pogg Ann 89 (1853): 211-33, 352-77. Repr. in WA 1 pp.475-519;<br />
Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber die Erhaltung der<br />
Kraft." In Pogg Ann 89 (1853): 568-579; Helmholtz, Hermann. "Erwiderung auf die<br />
Bemerkungen von Hrn. Clausius". In Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882,<br />
pp.76-93; Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber die<br />
Erhaltung der Kraft", zweite Notiz." In Pogg Ann 91 (1854): 601-604.<br />
5 For instance the "English (actually: British) controversy" in the Philosophical<br />
Magazine (1862-65). See: Helm, Georg. Die Energetik nach ihrer geschichtlichen<br />
Entwickelung . Leipzig: Veit, 1898. Pp.126-130; Lloyd, J.T. "Background to the Joule-Mayer<br />
Controversy." In Notes <strong>and</strong> Records of the R.S. 25 (1970): 211-25.<br />
6 See the interesting outlines in: C.Jungnickel,R.McCormmach, Intellectual Mastery<br />
of Nature , 2 vols, Chicago: Chicago U.P., 1986, vol 2, pp.211-53; Hiebert, Erwin. "The<br />
Energetics Controversy <strong>and</strong> the New Thermodynamics." In Duane H.D. Roller (ed).<br />
Perspectives in the History of Science <strong>and</strong> Technology . Norman: Oklahoma U.P., 1971.<br />
Pp.67-86.<br />
7 See expecially: Planck, Max. Das Prinzip der Erhaltung der Energie . Leipzig:<br />
Teubner, 1887. 2nd ed 1908; Helm, Georg. Die Lehre von der Energie, historisch-kritisch<br />
entwickelt . Leipzig, 1887; Helm, Georg. Die Energetik nach ihrer geschichtlichen<br />
Entwickelung . Leipzig: Veit, 1898; Haas, Arthur Erich. Die Entwicklungsgeschichte des<br />
Satzes von der Erhaltung der Kraft , Vienna: Hölder, 1909; but also: Mach, Ernst. Die<br />
Geschichte und die Würzel des Satzes von der Erhaltung der Arbeit . Prag, 1872. 2nd edition<br />
Leipzig,1909. Trans Philip E.B.Jourdain. History <strong>and</strong> Root of the Principle of the<br />
<strong>Conservation</strong> of <strong>Energy</strong> . Chicago: Open Court, 1911; Mach, Ernst. Die Mechanik in ihrer<br />
Entwickelung historisch-kritisch dargestellt . Prag, 1883. Trans. The Science of Mechanics .<br />
La Salle: Open Court, 1942; Mach, Ernst. Die Prinzipien der Wärmelehre, historisch-kritisch
The depth <strong>with</strong> which the literature was analysed is impressive. Not long<br />
afterwards, at the beginning of this century, comes the period of the<br />
philosophers 8, whose debates on the meaning of the principles of conservation<br />
are an indication of their lasting importance. Finally comes the time of the<br />
historians, who in the last thirty years have tried to give an account of this<br />
complex but fascinating scientific struggle, even if, unfortunately, often <strong>with</strong> a<br />
certain detachment, a lack of knowledge <strong>and</strong> acknowledgement, from the results<br />
of previous periods.<br />
Two more recent aspects of the historians' activity in the field seem<br />
promising: first, an improvement in the efforts to (re)discover <strong>and</strong> print so far<br />
unpublished sources preserved in the archives; as far as Helmholtz is concerned,<br />
this "industry" is well alive, produces relevant documents 9 <strong>and</strong> raises new<br />
expectations 10. Second, a historiographical problem shift: the focus on the<br />
emergence <strong>and</strong> development of theoretical physics (in Germany) 11 is offering new<br />
entwickelt , Trans. Principles of the Theory of Heat ,Historically <strong>and</strong> Critically Elucidated.<br />
Norwell, Mass.: Kluwer, 1986; Maxwell, James Clerk. The Theory of Heat , London, 1870;<br />
Maxwell, James Clerk. Matter <strong>and</strong> Motion , London, 1877; Ostwald, Wilhelm. Die Energie .<br />
Leipzig: Barth, 1908; Poincaré, Henry. La Science et l'Hypothèse . Paris, 1902. Rep Paris:<br />
Flammarion, 1968; Stallo,J.B. The Concepts <strong>and</strong> Theories of Modern Physics . London, 1882;<br />
Steward, Balfour.The <strong>Conservation</strong> of <strong>Energy</strong> .London, 1874; Rühlmann, Moritz. Vorträge<br />
Über Geschichte der technischen Mechanik und der theoretischen Maschinenlehre . Vols 2.<br />
Leipzig, 1881-1885.<br />
8 For instance: Meyerson, Emile. Identité et Realité , Paris, 1908; Cassirer, Ernst.<br />
Substanzbegriff und Funktionsbegriff, Berlin: B.Cassirer, 1910; 2nd ed 1923.<br />
9 Helmholtz, Hermann.Über die Erhaltung der Kraft . Christa Kirsten ed. Weinheim:<br />
Physik-Verlag, 1983 (a transcription of the penultimate manuscript); Kirsten, Christa, et al.,<br />
eds. Dokumente einer Freundschaft: Briefwechsel zwischen Hermann von Helmholtz und<br />
Emil du Bois-Reymond 1846-94. Berlin: Akademie-Verlag, 1986; Cahan, David. An Institute<br />
for an Empire. The Physikalisch-Technische Reichsanstalt. 1871-1918 . Cambridge:<br />
Cambridge U.P., 1989; Kremer, Richard. Letters of Hermann von Helmholtz to his Wife,<br />
1847-1859. Stuttgart: Steiner,1990.<br />
10 David Cahan. Letters of Hermann von Helmholtz to his Father . Forthcoming<br />
11 McCormmach, Russell. "Editor's Foreward" In HSPS 3 (1971) ix-xxiv; Olesko,<br />
Kathrin. "The Emergence of Theoretical Physics in Germany: Franz Neumann <strong>and</strong> the<br />
Königsberg School of Physics, 1830-1890." Dissertation. Cornell University, 1980;
<strong>and</strong> interesting results, despite the fact that the definition of "theoretical physics"<br />
is not <strong>with</strong>out difficulties. In my view the results achieved would be enhanced if a<br />
better definition could be reached 12, outlining in a precise way that theoretical<br />
physics is a discipline different both from experimental <strong>and</strong> from mathematical<br />
physics 13.<br />
In the light of these recent trends I propose that <strong>Helmholtz's</strong> Erhaltung 14<br />
be seen not in the light of a "discovery","simultaneous" or not, of a once <strong>and</strong> for<br />
all defined concept of "energy" or "principle of energy conservation", but as a<br />
central step in the process of the emergence of theoretical physics, for the explicit<br />
<strong>and</strong> sophisticated methodology outlined <strong>and</strong> the (neither experimental nor<br />
mathematical) results achieved. <strong>Helmholtz's</strong> theoretical effort resulted in a<br />
formulation of the principle of energy conservation based on the impossibility of<br />
perpetual motion <strong>and</strong> on the Newtonian model of forces depending only on<br />
positions. This allowed him to define two main forms of energy sharply split:<br />
potential <strong>and</strong> kinetic. At the same time <strong>Helmholtz's</strong> methodology made clear the<br />
distinction between theoretical <strong>and</strong> experimental physics <strong>and</strong> assessed a hierarchy<br />
of interacting levels in the structure of physical theories. This methodology is also<br />
a useful tool to analyse the different research programs that, based on the<br />
different approaches to energy problems of the middle of last century, grew in the<br />
following decades. In <strong>Helmholtz's</strong> long scientific activity the methodology played<br />
a more stable role than his specific 1847 approach to the principle of energy<br />
C.Jungnickel, R.McCormmach, Intellectual Mastery ; Wolff,Stefan."Clausius' way to the<br />
kinetic theory-the beginnings of theoretical physics in Germany." Forthcoming.<br />
12 C.Jungnickel <strong>and</strong> R.McCormmach's title is a quotation from <strong>Helmholtz's</strong><br />
autobiography, but a precise definition of the term "theoretical physics" widely used in the<br />
subtitles ("Theoretical Physics from Ohm to Einstein; vol 1: The Torch of Mathematics 1800-<br />
70; vol 2: "The Now Mighty Theoretical Physics 1870-1925") is lacking. References are made<br />
to Boltzmann (1895): "Even the formulation of this concept is not entirely <strong>with</strong>out difficulty,"<br />
Vol.1 p.XV; to Wien (1915): vol.2 p.XV. Wolff too, facing the same problem, relies on<br />
Boltzmann: n. 89.<br />
13 To my knowledge this problem is only, <strong>and</strong> briefly, pointed to in Kuhn,<br />
Thomas."Mathematical versus Experimental Traditions in the Development of Physical<br />
Science." In The Journal of Interdisciplinary History 7 (1976):1-31. Rep. in Kuhn Ess<br />
Tension ; see n.32. Pp. 64-5.<br />
14 Helmholtz, Hermann. Über die Erhaltung der Kraft, eine physikalische<br />
Abh<strong>and</strong>lung. Berlin: G.Reimer, 1847.
conservation. In fact the latter went through some serious modifications, also in<br />
relation to the debates <strong>with</strong> alternative formulations, ending <strong>with</strong> a refusal of this<br />
principle as the main research tool in physics. These very relevant conceptual<br />
shifts of Helmholtz on the energy problems should be seen as chess moves on the<br />
basis of a well defined methodology, lasting till the Einleitung to the Theoretical<br />
Lectures 15 of 1894.<br />
New trends in energy studies cannot avoid discussing Thomas Kuhn's paper on<br />
the "simultaneous discovery" of "energy conservation", published in 1959. Kuhn's<br />
paper is still unanimously defined as challenging, for the lack in contemporary<br />
historiography of a rival synthesis. Nevertheless analysing <strong>Helmholtz's</strong> Erhaltung<br />
, <strong>and</strong> the related primary <strong>and</strong> secundary literature, I started doubting of the<br />
correctness of some of Kuhn's main historical <strong>and</strong> historiographical claims.<br />
I will show in the last chapter that some major aspects of <strong>Helmholtz's</strong> Erhaltung<br />
escaped Kuhn's (<strong>and</strong> other historians') attention: 1) <strong>Helmholtz's</strong> methodological<br />
four level structure; 2) his demarcation between theoretical <strong>and</strong> experimental<br />
physics; 3) his overcoming of both the engineering <strong>and</strong> the mathematical<br />
approach to the work concept; 4) the lack of an experimental determination of a<br />
work equivalent of heat <strong>and</strong> the mistranslation of James Joule's values; 5) the<br />
difficult (<strong>and</strong> sometimes wrong) theory-experiment interplay in the application of<br />
the principle; 6) the formulation of a lasting methodology <strong>and</strong> of a non lasting<br />
conceptual model of energy.<br />
My work is meant as a contribution to: a) discussing the actual content of<br />
the Erhaltung ; b) pointing out the developments <strong>and</strong> modifications of the<br />
subsequent works of Helmholtz mainly in relation to the debates <strong>with</strong> Rudolf<br />
Clausius <strong>and</strong> Wilhelm Weber; c) raising the problem of a "return to the classics";<br />
d) discussing the recent historiographical debate.<br />
The theme of "energy" was a central one in <strong>Helmholtz's</strong> long scientific<br />
activity. When he decided to publish his own collected papers he gave "Zu Lehre<br />
von der Energie" the first place in the first volume 16. The three papers collected<br />
there belong to three different stages of <strong>Helmholtz's</strong> early researches on energy:<br />
the first, that is also the first of <strong>Helmholtz's</strong> famous Bericht , is the border-line<br />
between the early physiological researches <strong>and</strong> the physics of the Erhaltung . A<br />
15 Helmholtz, Hermann. Einleitung zu den Vorlesungen über Theoretische Physik .<br />
Arthur König <strong>and</strong> Carl Runge eds. Leipzig: Barth, 1903.<br />
1882.<br />
16 Helmholtz, Hermann. Wissenschaftliche Abh<strong>and</strong>lungen. 1st vol. Leipzig: Barth,
principle of correlation is expressed <strong>and</strong> various applications given, but the<br />
"energy" terms of the equations are expressed in units of heat <strong>and</strong> not work.<br />
The Erhaltung is the full expression of <strong>Helmholtz's</strong> ideas on theoretical<br />
physics <strong>and</strong> on energy. Its "Einleitung" is the clear <strong>and</strong> conscious indication of<br />
the methodological control Helmholtz had achieved on his own researches.<br />
Helmholtz outlined a four levels structure. Two basic physical hypotheses<br />
(impossibility of perpetual motion <strong>and</strong> central Newtonian forces) were placed at<br />
the first level, "the" principle of conservation at the second, empirical laws at the<br />
third <strong>and</strong> natural phenomena at the fourth. Moreover the two basic hypotheses,<br />
echoing various elements of Kantian philosophy, were not presented as self<br />
evident but as the result of a philosophical 'explanation'..<br />
In my view the need that Helmholtz felt to justify his own version of the<br />
conservation principle on higher grounds (on two physical hypotheses in turn<br />
justified on philosophical grounds) is a clear indication of <strong>Helmholtz's</strong><br />
consciousness of the possibility of alternative formulations of the principle itself.<br />
Helmholtz not only wanted to express a principle, but also to establish the<br />
framework <strong>and</strong> the rules following which principles could be formulated <strong>and</strong><br />
used. This is what makes <strong>Helmholtz's</strong> approach mark a major step in the<br />
emergence of theoretical physics, <strong>and</strong> shows that his version of the principle was<br />
not only the application of a (meta)physical assumption, but the application of a<br />
sophisticated methodology. The two physical assumptions (first level) are meant<br />
to bring together, not <strong>with</strong>out problems, two different but well known traditions<br />
in physics 17, <strong>and</strong> thus to offer secure grounding for the whole enterprise.<br />
Helmholtz believed that this was not enough <strong>and</strong> decided to justify the first level<br />
on more abstract grounds: he connected the principle of impossibility of perpetual<br />
motion <strong>with</strong> the principle of sufficient reason, a transcendental condition for the<br />
intelligibility of nature, <strong>and</strong> gave a conceptual explanation of the model of central<br />
forces in the Kantian style. Finally he hinted at an "empirical" principle of a cause<br />
effect relationship to be embedded in the formulation of the principle of<br />
conservation (second level). This principle, in turn, had to be compared <strong>with</strong><br />
existing empirical laws (third level) <strong>and</strong> to predict new ones, to eventually<br />
achieve the intelligibility of natural phenomena (fourth level).<br />
17 In this context I think that Elkana's suggestion of Helmholtz unifying Newtonian<br />
<strong>and</strong> analytic mechanics can be accepted: Elkana The Discovery ; but for a better<br />
characterization of trends in mechanics see: Grattan-Guinness, Ivor. "The varieties of<br />
mechanics by 1800", paper for the Hegel <strong>and</strong> Science Conference, typescript, 1989.
With reference to these four levels Helmholtz could draw an explicit<br />
distinction between theoretical physics (dealing <strong>with</strong> deductions from level two to<br />
three, that is <strong>with</strong> the applications of the principle to empirical laws) <strong>and</strong><br />
experimental physics (dealing <strong>with</strong> the inductions from level four to three, that is<br />
from natural phenomena to empirical laws). It is in my view important to remark<br />
that the dividing element between theoretical <strong>and</strong> experimental physics is not<br />
meant to be the use of mathematics. Thus theoretical physics was not at all<br />
identified <strong>with</strong> mathematical physics.<br />
In my opinion, <strong>with</strong> this Introduction, Helmholtz marks explicitly the<br />
emergence of theoretical physics, as a discipline distinct not only from<br />
experimental physics but also from mathematical physics. The most striking<br />
characteristic is <strong>Helmholtz's</strong> consciousness of the role of the four levels of the<br />
hierarchy, a consciousness that allowed him later to modify some specific<br />
elements of his scheme, leaving the methodological structure unchanged. The<br />
great <strong>and</strong> successful novelty is the stress on the interplay of the second <strong>and</strong> third<br />
level : since 1847 physical laws (level three) had to satisfy more <strong>and</strong> more not<br />
only experiments <strong>and</strong> natural phenomena (level four), but also theoretical<br />
principles (level two). These principles were also to be seen as heuristic tools to<br />
"discover" empirical laws: a complete new field of inquiry is open, theoretical<br />
physics, which is not primarily based on an extended use of mathematics. A basic<br />
characteristic of the principles is to be general, that is, to unify the different<br />
branches of physical knowledge.<br />
<strong>Helmholtz's</strong> methodological worries show also that he was obviously<br />
aware of the difficulties of his gigantic plan to unify natural sciences under one<br />
regulative principle. As I try to show below the Erhaltung is in fact "only" a plan<br />
that requires corroboration. If a discovery was made by Helmholtz in 1847, it<br />
was not that (of a specific formulation) of energy conservation, still a theoretical<br />
proposal lacking new experimental data <strong>and</strong> a secure mathematical grounding 18,<br />
but that of theoretical physics.<br />
In the Erhaltung an effort at showing the equivalence between the two<br />
main hypotheses (level one) is made in the first chapter, while the deduction from<br />
these hypotheses of <strong>Helmholtz's</strong> version of the principle of conservation (level<br />
two) is the object of the second chapter. The following four chapters deal <strong>with</strong><br />
the interactions between levels two <strong>and</strong> three, that is <strong>with</strong> the application of the<br />
18 See below the difficulties <strong>with</strong> the geometrical interpretation of the integral, in<br />
section 3 of the Erhaltung, <strong>and</strong> <strong>with</strong> the definition of selfpotential, in section 6.
principle to the existing empirical laws <strong>and</strong> <strong>with</strong> the attempt at deriving new<br />
empirical laws on theoretical grounds. In the Erhaltung no new experimental<br />
data (level four) are offered <strong>and</strong> the few regarding the mechanical equivalent of<br />
heat available at the time are criticised, also on the basis of a mistake on the<br />
conversions of the units of measurement, <strong>and</strong> disregarded.<br />
What Helmholtz really did in his Erhaltung was to lay the foundations of<br />
theoretical physics, through the conscious interplay of conceptual models,<br />
regulative principles, mathematical techniques <strong>and</strong> experimental results. His<br />
formulation of the energy principle was a specific application of this wideranging<br />
methodology <strong>and</strong> thus was meant to have the greatest number of possible<br />
applications.<br />
But <strong>Helmholtz's</strong> foundations of theoretical physics while being connected<br />
are not coextensive <strong>with</strong> his approach to energy conservation <strong>and</strong> will survive<br />
the latter. <strong>Helmholtz's</strong> methodology was in fact extraordinarily successful.<br />
<strong>Energy</strong> theory, that is theoretical physics applied to energy problems,<br />
changed the shape of physics : after 1847 physical laws no longer had only to<br />
face the challenge of experimental results, but also had to be judged on more<br />
theoretical grounds, in their relation <strong>with</strong> the conservation principle. But in turn<br />
the principle had, from the beginning, different formulations. From the point of<br />
view of <strong>Helmholtz's</strong> methodology, the other formulations of the principle of<br />
energy conservation can be seen as less methodologically sophisticated, but not<br />
less physically interesting. From 1847 on in physics the traditional theoryexperiment<br />
interplay was united <strong>with</strong> <strong>and</strong> often overcome by theory-principle<br />
interplay.<br />
Poggendorff's refusal of the paper for publication in the Annalen is an<br />
indication of the difficulty for experimental physicists to accept the new ideas,<br />
but it did not last long. An English version of the Erhaltung appeared in 1853<br />
<strong>and</strong> the third paper of the energy section of <strong>Helmholtz's</strong> collected works<br />
mentioned above reveals that, just a few years after Poggendorff's refusal, the<br />
controversy <strong>with</strong> Clausius (1852-54), which revealed the first deep differences<br />
between a theoretical <strong>and</strong> a more mathematical approach, was entirely published<br />
in the Annalen . Moreover in six reports (1855-59), written for the Fortschritte<br />
der Physik <strong>and</strong> dealing <strong>with</strong> the "Theorie der Wärme", Helmholtz discussed <strong>and</strong><br />
compared his own approach <strong>with</strong> an already wide literature: this might be<br />
considered the first history of energy conservation. Between 1862 <strong>and</strong> 1865, in<br />
the Philosophical Magazine , an international controversy on the "priority" of the<br />
"discovery" showed that the topic had become basic in scientific research.
Helmholtz himself, starting in 1854, was playing a major role in<br />
popularising the new view, but, surprisingly, almost hiding his initial conceptual<br />
model of central Newtonian forces <strong>and</strong> stressing instead the other assumption, the<br />
impossibility of perpetual motion. This shift, which again demonstrates<br />
<strong>Helmholtz's</strong> control of the various interacting elements, is in my view the result of<br />
Clausius' criticisms, but also of a growing interest in potential theory <strong>and</strong> in its<br />
applications, for instance in hydrodynamics (1858 <strong>and</strong> 1868) 19. Also relevant are<br />
the difficulties created for Newtonian forces by the growing field of<br />
electromagnetic phenomen. The debates of the seventies underline the<br />
methodological relevance of the 'energy principle', now a main tool for<br />
comparison of experimentally equivalent alternative theories, but at the same<br />
time outline the problem of identifying 'the' principle: various versions were, as<br />
usual, competing. <strong>Helmholtz's</strong> version reveals some limits of the original 1847<br />
assumptions.<br />
In these years Helmholtz faced the challenge of the nativists <strong>and</strong> the<br />
metaphysicians <strong>and</strong>, also as a result of his work on physiology <strong>and</strong> the<br />
foundations of geometry, shifted towards a more empirical side of his<br />
methodological approach on many issues. A strict adherence to Kantian<br />
categories <strong>and</strong> Kantian conceptual models was ab<strong>and</strong>oned, for instance the<br />
central Newtonian forces, while the methodological Kantian structure of the<br />
Erhaltung is preserved, in particular the need for a regulative principle that<br />
fulfils the "intelligibility of nature".<br />
In the last two decades of the nineteenth century the history of energy<br />
conservation entered a new phase: the differences between the original versions<br />
of the principle grew into different research programs. The mechanical view of<br />
nature <strong>and</strong> the belief in the reversibility of natural phenomena championed by<br />
Helmholtz were challenged by other, competing views. The energetist movement<br />
took off, <strong>with</strong> its attempts to overthrow the mechanical worldview, while the<br />
electromagnetic <strong>and</strong> the thermodynamic approach, both deeply connected <strong>with</strong><br />
energy problems, were actively pursued. Since energy conservation had become<br />
one of the basic chapters of physics, the champions of the different research<br />
programs dedicated a great deal of logical <strong>and</strong> historical analysis to an<br />
underst<strong>and</strong>ing <strong>and</strong> framing of the various contributions <strong>and</strong> developments of<br />
energy theory. Given that a specific version of the conservation principle played a<br />
great role in each research program, a series of works started to analyse in detail<br />
19 Helmholtz "Integrale" <strong>and</strong> " Flüssigkeitsbewegungen".
the origin <strong>and</strong> meaning of the principle, from different perspectives, of course.<br />
Helmholtz was a leader in the spread of theoretical physics in the second half of<br />
the century, <strong>and</strong> it is interesting to find out what was the judgement of his fellow<br />
scientists on his approach to the principle of energy conservation. At the turn of<br />
the century the great debates of the "now mighty theoretical physics" showed the<br />
progress achieved in this field: the mechanic, energetic, thermodynamic <strong>and</strong><br />
electromagnetic views of nature were the frameworks <strong>with</strong>in which basic studies<br />
on the history of energy conservation were written.<br />
But already in the eighties new interpretations of electromagnetic energy<br />
had opened the way to <strong>Helmholtz's</strong> shift towards Maxwell's theory of contiguous<br />
action <strong>and</strong> to the ab<strong>and</strong>onment of the traditional continental direct action at a<br />
distance approach (even if Helmholtz always translated Maxwell in his own<br />
terms of action at a distance plus dielectric). At the same time Helmholtz<br />
renounced the principle of energy conservation as the main basic tool in physical<br />
research <strong>and</strong> substituted it <strong>with</strong> the principle of least action (1886) 20. But after<br />
almost fifty years since the Erhaltung the basic aspects of his methodology were<br />
explicitly reasserted in a discussion of the foundations of theoretical physics 21.<br />
<strong>Helmholtz's</strong> latest results will have an influence almost exclusively on Hertz, but<br />
his continuous struggle shows a great methodological consistency. The<br />
"Einleitung" to the Lectures on Theoretical Physics of 1894 strongly resembles<br />
the methodology expressed in 1847.<br />
A few years later philosophers took an interest in the problems of the history <strong>and</strong><br />
foundations of energy conservation, <strong>and</strong> again Kantian influences were at issue.<br />
In recent times, towards the end of the neo-positivist trend historians again<br />
started to offer contributions on this important topic: perhaps it is time to try to<br />
put these latest efforts in historical perspective.<br />
From Physiology to Physics (1845-46)<br />
20 Helmholtz, Hermann. "Über die physikalische Bedeutung des Princips der kleinsten<br />
Wirkung." In Crelle's Journal 100 (1886) :137-166 <strong>and</strong> 213-222; Repr. in WA 3. Pp.203-248.<br />
21 Helmholtz Einleitung
In 1847 Helmholtz, a military surgeon in Potsdam, published two papers,<br />
both dedicated to energy conservation 22. Helmholtz had been a student of Müller<br />
in Berlin <strong>and</strong> was still closely in touch <strong>with</strong> some of his best pupils, namely<br />
DuBois-Reymond, Brucke <strong>and</strong> Ludwig. The latter together <strong>with</strong> some of Magnus'<br />
students, namely Gustav Karsten, Wilhelm Beetz, Wilhelm Heintz <strong>and</strong> Hermann<br />
Knoblauch, in 1845 founded the Physikalische Gesellschaft. The society started<br />
publishing the Fortschritte der Physik <strong>and</strong> <strong>Helmholtz's</strong> "Bericht" of 1847, written<br />
at DuBois's request, was the first of his numerous contributions to that journal.<br />
Physiology was, <strong>with</strong>out doubt, the professional <strong>and</strong> research context in which<br />
the two 1847 works on energy were written: <strong>Helmholtz's</strong> three works 23 published<br />
before 1847 <strong>and</strong> the two 24 in 1848 were dedicated to physiological problems 25.<br />
22 In the first volume of his Wissenschaftliche Abh<strong>and</strong>lungen of 1882 ( WA 1 )<br />
Helmholtz gave the first place to the section : "Zuhr Lehre von der Energie". Here he included<br />
three works: Helmholtz, Hermann. "Bericht über “die Theorie der physiologischen<br />
Wärmeerscheinungen“ betreffende Arbeiten aus dem Jahre 1845." In Fort d Ph im Jahre 1845<br />
1 (1847): 346-55; repr. in WA1 , 1882, Pp.3-11; Helmholtz, Erhaltung . Rep. in WA 1 , 1882,<br />
Pp.12-75; Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In<br />
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, Pp.76-93.<br />
23 Helmholtz, Hermann. "Ueber das Wesen der Fäulniss und Gährung". In<br />
Müller's Arch (1843): 453-62; repr.in WA 2, 1883, Pp.726-34; Helmholtz,<br />
Hermann. "Ueber den Stoffverbrauch bei der Muskelaction". In Müller's Arch<br />
(1845): 72-83; repr. in WA2 , 1883, Pp.735-44; Helmholtz, Hermann. "Wärme,<br />
Physiologisch". In Encyklopädisches Wörterbuch der medicinischen<br />
Wissenschaften, herausgegeben von Professoren der medicinischen Facultät zu<br />
Berlin. Vol.35, Berlin: Veit &Co,1846. Pp.523-67. Repr. in WA 2, 1883, Pp.680-<br />
725.<br />
24 Helmholtz, Hermann. "Ueber die Wärmeentwicklung bei der Muskelaction". In<br />
Müller's Arch (1848): 144-64; repr. in WA2, 1883, Pp.745-63. It had been presented on the<br />
12th of November 1847 to the Physikalische Gesellschaft; Helmholtz, Hermann. "Bericht über<br />
“die Theorie der physiologischen Wärmeerscheinungen“ betreffende Arbeiten aus dem Jahre<br />
1846". In Fort d Ph im Jahre 1846 2 (1848): 259-60.<br />
25 The three works of 1843, 1845, 1846 <strong>and</strong> the first of 1848 were published in<br />
physiological journals <strong>and</strong> in a medical encyclopedia <strong>and</strong> reprinted in the "Physiologie" section<br />
of W A 2. The two works of 1847 were addressed to physicists: the "Bericht" appeared in the
Whether physiological research, apart from being the context, had been<br />
also the root or one of the roots of <strong>Helmholtz's</strong> formulation of energy<br />
conservation, has been recently discussed at length 26. <strong>Helmholtz's</strong> own point of<br />
view, expressed in 1882 <strong>and</strong> 1892 27, is that his interest in energy conservation did<br />
not arise from empirical problems in physiology but from the inclination he had<br />
acquired from a very early age in favour of the principle of the impossibility of<br />
perpetual motion 28. Autobiographical reconstructions of scientists are often<br />
unreliable, but this antiempirical remark of Helmholtz was expressed at a time<br />
when he was stressing empirical elements in science 29 <strong>and</strong> thus deserves<br />
attention. In fact in the early eighties, after the controversies <strong>with</strong> Friederich<br />
Zöllner <strong>and</strong> Karl Eugen Dühring, Helmholtz underlined, at variance <strong>with</strong> his own<br />
previous judgements, the empirical components in the early formulations of the<br />
principle of conservation.<br />
Physiology thus, according to Helmholtz, offered only the battleground<br />
for an explanation of animal heat based on the principle of impossibility of<br />
perpetual motion <strong>and</strong> on the consequent refusal of vital forces.<br />
One of the problems of mid-century German physiology was in fact the<br />
acceptance or refusal of vital forces in the explanation of the origins of animal<br />
heat 30. Liebig played a great role in this debate: already in 1841 he asserted a<br />
Fortschritte der Physik, the Erhaltung was presented to the Berlin Physikalische Gesellschaft<br />
on the 23rd of July <strong>and</strong> submitted to the Annalen der Physik.<br />
26 R.Kremer in his perceptive dissertation: "The Thermodynamics of Life <strong>and</strong><br />
Experimental Physiology, 1770-1880." Harvard University, 1984, Pp.190-3 contrasts the<br />
"st<strong>and</strong>ard" view that researches in energy conservation were motivated by physiological<br />
problems.<br />
27 Helmholtz, Hermann. "Über die Erhaltung der Kraft" in WA1 , 1882, P.74, <strong>and</strong><br />
Helmholtz, Hermann. Autobiographical Sketch , Pp.10-12.<br />
28 Koenigsberger accepted this approach: Koenigsberger, Leo. Hermann von<br />
Helmholtz. Tr. by Frances A. Welby. Oxford: Clarendon, 1906. P.8 <strong>and</strong> pp.25-6; Kremer gives<br />
a different interpretation of <strong>Helmholtz's</strong> remarks: he is more interested in denying the<br />
relevance of vitalism in physiological debates than in the role ot the principle of impossibility<br />
of perpetual motion in <strong>Helmholtz's</strong> work; see:"Therm of Life" pp.237-238.<br />
29 Helmholtz "Über die Erhaltung der Kraft" in WA1 , 1882, Pp.71-4 <strong>and</strong> "Robert<br />
Mayer's Priorität". In Vorträge und Reden. 2 Vols. Braunschweig: Vieweg, 1884.<br />
30 On the relevance of the problem see: Lenoir, Timothy. The Strategy of Life:<br />
Teleology <strong>and</strong> Mechanics in Nineteenth Century German Biology . Dordrecht <strong>and</strong> Boston:
principle of correlation of forces, that is of conversion <strong>with</strong> constant coefficients:<br />
" From nothing no force can be generated..." 31 <strong>and</strong> in 1842, in his Tierchemie ,<br />
refused the idea that vital forces could generate animal heat 32.<br />
Helmholtz was greatly influenced in his first physiological researches by<br />
Liebig's approach 33, even if he eventually rejected it 34. <strong>Helmholtz's</strong> first two<br />
works had controversial results 35, nevertheless in 1845, following Liebig, he was<br />
explicit on the possibility of a common origin of mechanical forces <strong>and</strong> heat<br />
produced by an organism 36 <strong>and</strong> wondered whether this origin could be entirely<br />
attributed to metabolism, thus avoiding vital forces.<br />
Reidel, 1982. Pp. 195-6, 215-7, 230; his approach differs from Kremer's, see n.26. See also<br />
Olesko & Holmes "Experiment" P.12.<br />
31J.Liebig: Chemische Briefe, tenth letter in the 1845 edition; twelfth letter in the<br />
third edition (Heidelberg 1851) at pp. 116-118. First printed in the supplement to Allg.Ztg. 30<br />
Sept 1841. This is a famous passage: quoted in Helm Energetik P 10; Haas Entwickl P.57;<br />
Kuhn Sim Disc P.95.<br />
32 See Kremer: "Therm of Life", Pp.204-9.<br />
33 Kremer even asserts that "Every physiological question Helmholtz considered<br />
before 1847 had been thoroughly defined by Liebig": "Therm of Life" P.238. Liebig is often<br />
credited <strong>with</strong> being a pioneer in the history of energy conservation. See Planck Princip P.33;<br />
Helm Energetik P.10 ; Haas Entwickl P.57; more recently Kuhn Sim Disc P.68; Kremer<br />
"Therm of Life" Pp.198-215.<br />
34 Lenoir Strategy P.196.<br />
35 The first one, on fermentation <strong>and</strong> putrefaction, was meant to support Liebig's<br />
antivitalist position. For Koenigsberger H v H P.27 <strong>and</strong> Kremer ibid Pp. 239-40 the paper<br />
raised confusing conclusions, while for Lenoir The Strategy P. 197 <strong>and</strong> Yamaguchi:<br />
Yamaguchi, Chuhei. "On the Formation of <strong>Helmholtz's</strong> View of Life Process in His Studies of<br />
Fermentation <strong>and</strong> Muscle Action - In Relation to His Discovery of the Law of <strong>Conservation</strong> of<br />
<strong>Energy</strong>." In Historia Scientiarum 25 (1983) : 29-37,was important, stimulating <strong>and</strong> also a<br />
good device for further research. In his second work, on metabolism during muscular activity,<br />
Helmholtz was inconclusive as far as metabolism was concerned, also for he lacked an exact<br />
relation between the muscular action <strong>and</strong> the heat developed. See Koenigsberger H v H P.32,<br />
Kremer "Therm of Life" P.243, Lenoir Strategy P.202.See also Olesko & Holmes<br />
"Experiment" P.1.<br />
36 "whether or not the mechanical force <strong>and</strong> the heat produced in an organism could<br />
result entirely from his own metabolism" or whether part of the animal heat should be
In a discussion on the origins of animal heat 37 of 1846 some elements of<br />
the methodological strategy of the Erhaltung can already be identified.<br />
Helmholtz accepted Liebig's principle of force correlation <strong>and</strong> his theory<br />
on the chemical origin of heat, but asserted that two important points needed to<br />
be clarified for a satisfactory application of the principle: the conceptual model of<br />
heat <strong>and</strong> the definition of the heat equivalents which are utilized in the<br />
correlation. Liebig's theoretical determinations in fact did not agree <strong>with</strong><br />
experimental results.<br />
To question the conceptual model of heat as caloric might appear<br />
counterproductive in a discussion on the origin of animal heat: Helmholtz himself<br />
pointed out that this model was very useful to refute vital forces. In fact the<br />
conservation of matter assured that the amount of (latent) heat ingested was the<br />
same as that emitted by the living bodies. Even stronger support was given by<br />
German Ivanovich Hess's law asserting that in chemical transformations the order<br />
of the intermediate steps had no influence for the final emission of heat. Caloric<br />
was a definite, conserved quantity. But in a strategy that wanted to take into<br />
account the progress of all the sciences, <strong>and</strong> particularly of the physical ones, this<br />
model had to be changed: the recent identification of the thermal radiation <strong>with</strong><br />
light, <strong>and</strong> the generation of heat that could not be ascribed to the liberation of<br />
latent heat, for instance in electrical processes, compelled to accept the idea of<br />
heat as movement 38.<br />
The new model was not <strong>with</strong>out disadvantages: it added some problems<br />
to the refusal of vital forces. The total amount of heat was no longer considered<br />
constant <strong>and</strong> the production of heat through the action of forces was now<br />
admitted as possible. In principle vital forces could be considered among the<br />
forces producing heat 39. To deny this <strong>and</strong> to reassert the principle of impossibility<br />
of perpetual motion made compelling the solution of the second problem outlined<br />
above: the redefinition of the heat equivalents.<br />
attributed to the action of a vital force, specific to the organic life: "Muskelaction" WA 2<br />
P.735; see also Koenigsberger H v H P. 31 <strong>and</strong> Kremer "Therm of Life" Pp.240-1.<br />
37 Helmholtz:"Wärme, Physiologisch"; WA 2 Pp.695-700.<br />
38 For the role of the undulatory theory of heat see: Brush Kind of Motion .<br />
39 "to the physiologists that connect the essence of life <strong>with</strong> this very<br />
incomprehensibility, we have nothing to contrappose from a theoretical point of view";<br />
Helmholtz W A 2 P.700.
The amount of animal heat predicted by Liebig was in fact smaller then<br />
the one measured by Piere Louis Dulong <strong>and</strong> Cesar-Mansuète Despretz. Thus the<br />
difference might have been explained through the effect of vital forces. Helmholtz<br />
wanted to eliminate the discrepancy between theoretical predictions (Liebig) <strong>and</strong><br />
experimental results (Dulong <strong>and</strong> Despretz) in order to eliminate any role for the<br />
vital forces. He tried to achieve that through a theoretical reformulation of the<br />
terms of both sides of the equation relating the heat ingested <strong>with</strong> the one emitted<br />
by living bodies. Helmholtz proposed first that the heat of the "Ingesta" be<br />
considered no more as the one resulting from the oxidation of the elements of the<br />
food, but instead as the one resulting from the oxidation of the compound<br />
themselves of the food. The last one was supposed to be greater. Second<br />
Helmholtz proposed that the heat developed in the animals be considered not<br />
only as the one produced in the respiratory organs 40, but that also the heat<br />
produced in the blood <strong>and</strong> tissues should be taken in account 41. With this new<br />
elements, according to Helmholtz, theoretical predictions would satisfy<br />
experimental data <strong>and</strong> the acceptance of the new model of heat would leave no<br />
room for the vital forces. Still the experimental corroborations were highly<br />
problematic 42.<br />
I want to note that the mechanical equivalent of heat was<br />
discussed 43, even if a determination of the equivalent itself was lacking <strong>and</strong> thus<br />
a component was lacking in the theoretical energy balance put forward, namely<br />
the work done by the animals 44. Thus in my view Kuhn's assertion that in1845-6<br />
"Helmholtz fails to notice that body heat may be expended in mechanical work" 45<br />
requires qualifications: it does not imply that "<strong>Helmholtz's</strong> conservation ideas<br />
were not complete till 1847". In fact, already in his "Muskelaction", Helmholtz<br />
asserted that the problem was "whether or not the mechanical force <strong>and</strong> the heat<br />
produced in an organism could result entirely from its own metabolism" 46 <strong>and</strong> in<br />
n.148 p.248.<br />
40 For Kremer this does not correspond to Liebig's conception: "Therm. of Life"<br />
41 Lenoir Strat of Life P.204<br />
42 Kremer: "Therm of Life" P.251<br />
43 Helmholtz: WA 2 Pp.699-700.<br />
44 Kremer asserts that Helmholtz never succeeded in bringing heat <strong>and</strong> work<br />
operationally together into physiological research: Kremer "Therm of Life" P.238.<br />
45 Kuhn Sim Disc p.95 n 68.<br />
46 see above n.36.
his "Wärme" that one of the differences between the kinetic <strong>and</strong> the caloric<br />
theory of heat is "the determination of the equivalent of the heat that can be<br />
produced through a given quantity of a mechanical or electric force" 47. A more<br />
detailed discussion, as shown below, is in the "Bericht". It is true that Helmholtz<br />
did not provide in the first papers a mechanical equivalent of heat, but not for the<br />
lack of adequate conceptualization. He lacked experimental reliable data, a<br />
problem still present in the Erhaltung . Helmholtz himself, in the last section of<br />
the Erhaltung , explained that given that the amount of work produced by<br />
animals compared to the heat is small 48, it can be neglected <strong>and</strong> the problem of<br />
conservation of force in physiology is reduced to the problem whether the<br />
combustion <strong>and</strong> the transposition of food can produce the same amount of heat<br />
produced by the animals. Helmholtz added that the results of his own work in the<br />
"Wärme" <strong>and</strong> the "Bericht", compared <strong>with</strong> Dulong <strong>and</strong> Despretz's<br />
measurements, allowed a positive answer, at least approximately.<br />
Thus Kuhn's remarks, prompted by his desire to stress presumed<br />
influences of Natürphilosophie on Helmholtz, seem dubious. Even contradictory<br />
if compared <strong>with</strong> his other remarks 49, written to deny that Helmholtz had been<br />
influenced by the concepts of conservation present in the tradition of analytical<br />
mechanics, that <strong>Helmholtz's</strong> conservation ideas pre-existed his 1842 reading of<br />
D.Bernoulli.<br />
A great step forward in the elaboration of the methodological strategy was<br />
achieved in October 1846 50 in the first "Bericht" written for the Fortschritte,<br />
where it was published in 1847. The word "methodological" must be stressed<br />
because despite Koenigsberger's assertion 51 that Helmholtz at the time was<br />
involved in detailed experimentations, no new experimental result whatever is in<br />
this paper. The relevant feature is the extension of the correlation principle from<br />
physiology to various branches of physics <strong>and</strong> chemistry <strong>and</strong> thus this was<br />
<strong>Helmholtz's</strong> first attempt at a general application of his methodology. Without<br />
47 Helmholtz "Wärme" WA2 Pp.699-700.<br />
48 See also the views expressed in the "Wärme" quoted in Olesko & Holmes<br />
"Experiment" Pp.21-2 n.70.: "(animal heat) is by far the greatest part of the Kraftequivalent".<br />
49 Kuhn Sim Disc p.97 n.76.<br />
50 Koenigsberger H v H P.34.<br />
51 ibidem pp.34-5.
doubts this "Bericht" is a relevant work, even if it has not yet received the<br />
attention it deserves 52.<br />
Summarising the problem discussed in his 1846 paper, Helmholtz<br />
explicitly asserted that heat cannot originate out of nothing (nicht aus nichts ) <strong>and</strong><br />
stated a principle of correlation <strong>with</strong> Liebig's words: Das Princip von der<br />
Constanz des Kraftaquivalents bei Erregung einer Naturkraft durch eine <strong>and</strong>ere<br />
53 (the principle of the constancy of the force-equivalents for the stimulation of<br />
one force of nature through another). But Helmholtz, while retaining the principle<br />
of correlation, criticised its application to animal heat. As already mentioned<br />
Liebig's solution did not correspond to the calorimetric results of Dulong <strong>and</strong> of<br />
Despretz: the direct measurement of the combustion heat of the quantities of<br />
hydrogen <strong>and</strong> carbonium corresponding to the food ingested was only from 70 to<br />
90% of the heat provided by the animals.<br />
But the correlation principle should not be confined to the problem of<br />
animal heat. It is in fact based on the impossibility of perpetual motion, that<br />
"logically is completely justified", <strong>and</strong> has already been applied in the<br />
"mathematical theories" of Carnot-Clapeyron (still based on the conceptual model<br />
of the caloric) <strong>and</strong> of Franz.Neumann (based on the concept of electrodynamic<br />
potential). Nevertheless, Helmholtz remarked, the principle has not yet been<br />
expressed in complete form nor experimentally verified, despite a complete<br />
corroboration from the experimental results already achieved. Helmholtz saw<br />
correlation as a more sophisticated expression of the principle of impossibility of<br />
perpetual motion <strong>and</strong> immediately utilised it, offering a series of energy balances<br />
based on the new model of heat as movement. On the basis of both the<br />
"constancy of the force-equivalents" 54 principle <strong>and</strong> of the mechanical model of<br />
heat 55, it must result that "mechanical, chemical <strong>and</strong> electric forces can always<br />
few lines.<br />
52 For instance Lenoir does not deal <strong>with</strong> this paper <strong>and</strong> Kremer dedicates to it only<br />
53 Helmholtz W A 1 P.6. See also P.4 (reference is here to Liebig's 1845 "Ueber die<br />
tierische Wärme").<br />
54 Helmholtz W A 1 P.6<br />
55"But at present the material theory of heat is no more acceptable, but should be<br />
substituted <strong>with</strong> a kinetic one, for we see that heat originates from mechanical forces either<br />
directly, i.e. through the friction of solid bodies against solid bodies <strong>and</strong> of fluids against<br />
solids, or indirectly through electrical currents,from the movement of magnets <strong>and</strong> from
generate a determined equivalent of heat, however complicated the transition<br />
from one force to the other" 56. Helmholtz asserted that the empirical evidence<br />
was not very large but still he wanted to offer specific theoretical applications of<br />
the principle to the heat produced through mechanical, chemical, electrolytic <strong>and</strong><br />
electrostatic forces. The case of animal heat was now only the last of five<br />
applications of a principle that was becoming more <strong>and</strong> more general.<br />
A great difficulty immediately appeared: the most important balance, the<br />
one between heat <strong>and</strong> work, could not be written for the lack of a mechanical<br />
equivalent of heat; in fact the values offered by the theories of Carnot-Clapeyron<br />
<strong>and</strong> Holtzmann could not be accepted, being based on the caloric model <strong>and</strong> only<br />
referred to the propagation <strong>and</strong> not to the production of heat 57. Helmholtz, despite<br />
having clearly focussed the problem, lacks experimental data. In October 1846 he<br />
still did not know of Mayer's nor of Joule's work: "no experiment can be taken in<br />
account for the mechanical forces" 58. Thus all the other balances were written as<br />
equivalences based on heat units <strong>and</strong> not on work units. In the "Bericht" heat <strong>and</strong><br />
not work was the unity of measurement common to all the natural phenomena<br />
considered, not a minor difference <strong>with</strong> the subsequent Erhaltung.<br />
In the analysis of chemical transformations Hess' law of the constancy of<br />
the production of heat, whichever the intermediate stages of the reaction, was<br />
acknowledged. Here appeared the first instance of the identification of latent heat<br />
<strong>with</strong> the thermal equivalent that was to play a role in the subsequent discussion of<br />
animal heat.<br />
For the electrolytic currents the heat developed in the circuit must be<br />
equivalent to the electrochemical transformations in the galvanic chain (battery),<br />
independently of their order. The heat in the circuit could be calculated through<br />
Georg Simon Ohm's <strong>and</strong> Emily Christianovic Lenz's laws (Joule was not<br />
mentioned):<br />
frictional electricity, where a release of latent heat is inconceivable". Helmholtz: W A 1 Pp.6-<br />
7.<br />
56 Helmholtz W A 1 P.7<br />
57 It is interesting to remark that instead in the Erhaltung these data were utilised,<br />
that Clausius criticised this "improper" use, that Helmholtz accepted the criticisms, that<br />
Truesdell denies the validity of these criticisms. See below.<br />
58 Helmholtz WA 1 P.7.
where H is the heat, J the intensity of the electric current, W the total<br />
resistance of the circuit; on the other side, through Faraday's electrolytic law, we<br />
have:<br />
H=AC<br />
where A is the electrical "difference" 59 of the metals involved <strong>and</strong> C is the<br />
quantity of atoms "consumed" (atoms that underwent a process of oxidation <strong>and</strong><br />
reduction). According to the principle of equivalence, the heat produced in the<br />
circuit must be equivalent to the one that could be produced through the<br />
electrochemical transformations in the cells.<br />
For static electricity Helmholtz stated, in a couple of lines, that the<br />
production of heat through an electric discharge follows from Pieter Riess's<br />
principles; thus a balance was established between the resulting heat on one side<br />
<strong>and</strong> the product of the quantity of electricity by the electrical density (a still<br />
Voltian term for what was later identified <strong>with</strong> the tension or the difference of<br />
potential) on the other. <strong>Helmholtz's</strong> terminology in 1846 was still influenced by<br />
Volta's idea of tension <strong>and</strong> far from potential theory.<br />
Finally discussing animal heat, Helmholtz identified the latent heat of<br />
chemical reactions <strong>with</strong> the thermal equivalent that could be produced in further<br />
reactions. The energy balance must hold between the latent heat of the "Ingesta"<br />
on one side <strong>and</strong> the heat "provided by the animals" plus the latent heat of the<br />
"Egesta" 60 on the other side. The great contribution of Helmholtz is that the<br />
equivalent at the first side of the balance is no longer the "heat of combustion of<br />
hydrogen <strong>and</strong> carbonium but that of the food" 61. Through this reformulation <strong>and</strong><br />
the modification of the respiratory theory, Helmholtz claimed to have achieved a<br />
refutation of vital forces while being in agreement <strong>with</strong> Dulong's <strong>and</strong> Despretz's<br />
experimental results, <strong>and</strong> thus to have overcome Liebig's difficulties.<br />
It is evident that a new methodology had been acquired <strong>and</strong> that<br />
Helmholtz was aware of its great generality. The "Bericht" is in fact the border<br />
line between physiology <strong>and</strong> energy conservation; it is relevant to note that in<br />
many respects the methodology of the "Bericht" is very close to the one of the<br />
subsequent Erhaltung : to enunciate a principle, a conceptual model of the<br />
quantities involved, to express an equation between the energy terms <strong>and</strong> then to<br />
compare it <strong>with</strong> the empirical laws. There are some important differences : the<br />
59 Helmholtz WA 1 P.7<br />
60 Helmholtz WA 1 P.8<br />
61 Planck Princip P.34.
"Bericht", despite the application of the principle to an analysis of some physicochemical<br />
laws, is still largely dedicated to physiology. In the much longer<br />
Erhaltung , instead, physiology is confined to a few lines at the end of the last<br />
chapter. In the "Bericht" the equivalence principle based on the impossibility of<br />
perpetual motion (but also on the opposite impossibility of destroying motion:<br />
nothing can be created <strong>and</strong> nothing can be destroyed; a conservation of the<br />
coefficients of correlation applies) is present together <strong>with</strong> a model for many<br />
equivalents (the terms of the energy balance), but the equivalence principle, i.e.<br />
the correlation principle applied in the "Bericht" is very different from the<br />
mechanical principle of conservation of energy expressed in the Erhaltung. The<br />
equivalence principle is much closer to the ideas of Mayer <strong>and</strong> Joule, despite the<br />
fact that in 1846 Helmholtz did not know of their works (none of them was cited<br />
in the "Bericht"). In fact it only asserts the numerical equivalence of the effects<br />
involved <strong>and</strong> does not imply the assumption of central Newtonian forces <strong>and</strong> that<br />
every effect must have a mechanical interpretation in terms of potential <strong>and</strong><br />
kinetic energy 62. A final relevant point is that in the "Bericht" Helmholtz did not<br />
discuss the specific determinations of the mechanical equivalent of heat, despite<br />
his acceptance of the mechanical theory.<br />
In 1847 Helmholtz, while writing the Erhaltung, worked out a sixth<br />
paper 63 again dedicated to physiological problems, later to be reprinted in the<br />
"Physiologie" section of his collected papers..<br />
Here Helmholtz finally tried to link the problem of animal heat <strong>with</strong> that<br />
of the mechanical force produced by muscle action. Relevant to his purposes was<br />
to demonstrate that heat is produced in the muscle itself. He devised a very<br />
sensible thermocouple which, linked to an astatic galvanometer <strong>and</strong> a magnifying<br />
coil, could detect differences of temperature in the range of one thous<strong>and</strong>th of a<br />
degree centigrade. Through thorough experiments on frogs' legs Helmholtz was<br />
able to find evidence that heat is generated directly in the muscle tissue, that its<br />
origins are due to chemical processes <strong>and</strong> that production of heat in the nerves is<br />
negligible (<strong>and</strong> thus vital force could be disposed of on empirical grounds). The<br />
role of this experimental research on the sources of animal heat, carried forward<br />
together <strong>with</strong> <strong>and</strong> immediately after the writing of the Erhaltung, is very<br />
62Thus I cannot agree <strong>with</strong> Lenoir's assertion:"the physiology of muscle action laid<br />
before Helmholtz all the elements of conservation of energy" Lenoir Strat of Life P.211.<br />
63 Helmholtz "Wärmeentwicklung". A detailed analysis is given in Olesko & Holmes<br />
"Experiment" Sect 6.
important also for <strong>Helmholtz's</strong> underst<strong>and</strong>ing of energy conservation: it was in<br />
fact the only experimental research in this field that he made. <strong>Helmholtz's</strong><br />
underst<strong>and</strong>ing <strong>and</strong> evaluation of the mechanical equivalent of heat is probably to<br />
be connected <strong>with</strong> this very research 64.<br />
In the next section I will show that the entire Erhaltung only offered a<br />
theoretical reinterpretation of known results, but no new experiments. In my view<br />
physiology could not <strong>and</strong> did not provide a key to <strong>Helmholtz's</strong> conservation of<br />
"force": it did not offer theoretical nor experimental evidence for the<br />
establishment of the principle of force correlation (the refusal of vital force was,<br />
rather, based on an already accepted impossibility of perpetual motion).<br />
Nevertheless it did provide Helmholtz <strong>with</strong> an interesting battleground <strong>and</strong> his<br />
only original experimental data 65.<br />
The Erhaltung <strong>and</strong> its two different conceptual roots: the<br />
impossibility of perpetual motion <strong>and</strong> the Newtonian forces<br />
hypothesis.(1847)<br />
From October 1846 to July 1847 Helmholtz, not distracted but rather<br />
inspired by his love <strong>and</strong> engagement <strong>with</strong> Olga von Velten 66, worked hard at the<br />
Erhaltung. But we already know that this was not his only commitment: at the<br />
same time he was also involved <strong>with</strong> experiments on the heat produced during<br />
muscular action. Koenigsberger asserts that in the first quarter of 1847 Helmholtz<br />
had formulated his ideas on energy conservation <strong>and</strong> had tested them <strong>with</strong><br />
64 This point is made by Lenoir ibid. P.211, even if details are not given. This<br />
interpretation offers a clue to Koenigsberger's already recalled claims of deep experimentation<br />
carried forward by Helmholtz in 1846-47. See also Kremer ibid.P.244. Olesko & Holmes<br />
"Experiment" P.34.<br />
65 See also Koenigsberger, Leo. "The Investigations of Hermann von Helmholtz on<br />
The Fundamental Principles of Mathematics <strong>and</strong> Mechanics." In Annual Report of the Smith<br />
Inst to July 1986. Washington 1898. Pp.93-124. P.101.<br />
66 See the dedication, afterwards <strong>with</strong>drawn, on the manuscript of the Erhaltung:<br />
Helmholtz, Hermann.Über die Erhaltung der Kraft . Christa Kirsten ed. Weinheim: Physik-<br />
Verlag, 1983, P.14; see also Koenigsberger H v H Pp.35-7.
experiments in the most disparate branches of physiology <strong>and</strong> physics 67. This<br />
claim is difficult to accept: actually the Erhaltung, as I am about to show, does<br />
not have a specific experimental character, nor does it offer new experimental<br />
results. The only possible interpretation of Koenigsberger's claim is to relate all<br />
of <strong>Helmholtz's</strong> experimental activity of the period to the parallel researches on<br />
heat <strong>and</strong> muscular action, that is to the preparation of his sixth paper.<br />
The 23rd of July 1847 the Erhaltung was presented, <strong>with</strong> great success,<br />
at the Physikalische Gesellschaft. However, Magnus' <strong>and</strong> Poggendorff's<br />
judgements were not so warm <strong>and</strong> publication in the Annalen was denied. The<br />
essay was finally published at Reimer.<br />
Reimer's edition 68 of the Erhaltung consists of 1) an Introduction, of<br />
methodological <strong>and</strong> philosophical character, <strong>and</strong> six chapters. The first two<br />
67 Koenigsberger: ibid.P.37<br />
68 Different editions <strong>and</strong> translations of the Erhaltung have been published: A) the<br />
transcription of the penultimate manuscript: Helmholtz, Hermann.Über die Erhaltung der<br />
Kraft . Christa Kirsten ed. Weinheim: Physik-Verlag, 1983; B) Reimer's edition: Helmholtz,<br />
Hermann. Über die Erhaltung der Kraft, eine physikalische Abh<strong>and</strong>lung . Berlin: G.Reimer,<br />
1847; C) the "historical" translation of Tyndall: Helmholtz, Hermann. "On the <strong>Conservation</strong> of<br />
Force; a Physical Memoir." Trans.John Tyndall. In Scientific Memoirs-Natural Philosophy,<br />
Tyndall <strong>and</strong> Francis (eds.), vol. I, p.II, London,1853:114-162; D) a French translation,<br />
probably prompted by Clausius (see Wolff's "Clausius"): Helmholtz, Hermann. Mémoire sur la<br />
conservation de la force précédé d'un exposé élémentaire de la transformation des forces<br />
naturelles, trans L.Pérard. Paris: V.Masson et Fils, 1869; E) <strong>Helmholtz's</strong> own reprint in the W<br />
A 1 of 1882. This edition includes some footnotes <strong>and</strong> six appendix; moreover the only final<br />
note of Reimer's edition is included in the text: Helmholtz, Hermann. "Über die Erhaltung der<br />
Kraft" in WA1 , 1882, pp.12-75; F) the reprint in Ostwald's series. Here the 1882 footnotes do<br />
not appear: Helmholtz, Hermann. Über die Erhaltung der Kraft, eine physikalische<br />
Abh<strong>and</strong>lung . In Ostwalds Klassiker der Exacten Wissenschaften . Nr.1. Leipzig:<br />
W.Engelmann, 1889; G) an Italian translation, accurate but <strong>with</strong>out the 1882 footnotes:<br />
Helmholtz, Hermann. "Sulla Conservazione della Forza." In Opere . V.Cappelletti ed. Torino:<br />
UTET, 1967 : 49-116; H) a new English translation, in my view controversial: Helmholtz,<br />
Hermann. "The <strong>Conservation</strong> of Force: a Physical Memoir." In Selected Writings of Hermann<br />
von Helmholtz . R.Kahl ed. Middletown: Wesleyan U.P., 1971: 3-55; I) a third English<br />
translation, whiggish (<strong>Conservation</strong> of <strong>Energy</strong> ) but <strong>with</strong> comments: Helmholtz, Hermann.<br />
"On the <strong>Conservation</strong> of <strong>Energy</strong>." In Applications of <strong>Energy</strong>: Nineteenth Century . R.Bruce<br />
Lindsay ed. Stroudsburg: Dowden Hutchinson & Ross, 1976, Pp. 7-31 <strong>and</strong> 226-243.
chapters, 2)"The Principle of the <strong>Conservation</strong> of Living Force " <strong>and</strong> 3)"The<br />
Principle of <strong>Conservation</strong> of Force", are dedicated to formulating the principle;<br />
the following four chapters are dedicated to the applications of the principle in<br />
the fields of 4) Mechanics, 5) Thermology, 6) Electrostatics <strong>and</strong> Galvanism, 7)<br />
Magnetism <strong>and</strong> Electromagnetism, respectively.<br />
1 The "Einleitung": intelligibility of nature <strong>and</strong> conceptual explanation<br />
In February 1847 Helmholtz sent a sketch of the Erhaltung 's Introduction<br />
to DuBois: it was immediately praised as "an historical document of great<br />
scientific import for all time" 69. This apparently over-enthusiastic judgement<br />
proved to be correct: almost fifty years later Helmholtz was to apply the same<br />
concepts in the Introduction to his series of Lectures in Theoretical Physics ,<br />
ninety years later Einstein <strong>and</strong> Infeld quoted it 70 as the paradigm of the<br />
mechanical conception of nature, <strong>and</strong> today it still is the object of methodological<br />
debates 71.<br />
The history of the Introduction is complicated: when the Erhaltung was<br />
presented the 23rd of July at the Physikalische Gesellschaft <strong>and</strong> when sent to<br />
Magnus in the hope of publication in the Poggendorff's Annalen, the Introduction<br />
had been dropped. After Poggendorff's refusal, at DuBois' request, it was restored<br />
altered in "certain parts" 72 when the essay was sent to Reimer to be published.<br />
In my view these modifications can be identified <strong>with</strong> the addition of what<br />
is now the first paragraph of the Introduction: they represent an opening<br />
paragraph in which the plan of the Erhaltung is sharply summarised, <strong>and</strong> are of<br />
extraordinary relevance for an underst<strong>and</strong>ing of <strong>Helmholtz's</strong> approach.<br />
69 Koenigsberger : H v H P.37.<br />
70 Einstein, Albert <strong>and</strong> Infeld, Leopold. The Evolution of Physics. New York: Simon<br />
& Schuster, 1938; chapt.1, sect 9: The philosophical background.<br />
71 Turner, Steven. "Hermann von Helmholtz." In DSB 6, 1973. Pp.241-53; Heimann,<br />
Peter. "Helmholtz <strong>and</strong> Kant: The Metaphysical Foundations of 'Über die Erhaltung der Kraft'<br />
." In SHPS 5 (1974) : 205-38; Galaty, David. "The Philosophical Basis of Mid-19th Century<br />
German Reductionism". In Journal for the History of Medicine <strong>and</strong> Allied Sciences 29<br />
(1974): 295-316; Cohen,Robert <strong>and</strong> Elkana, Yehuda. "Introduction". In H.v.Helmholtz .<br />
Epistemological Writings . Boston: Reidel, 1977. Pp.IX-XXVIII. .<br />
72 Koenigsberger : H v H. P.38.
The premise reveals that the structure of the Erhaltung is based on four<br />
relevant methodological layers 73: a) to establish two physical assumptions<br />
("physikalischen Voraussetzung": central Newtonian forces <strong>and</strong> impossibility of<br />
perpetual motion) <strong>and</strong> their equivalence 74; b) to derive from them as a<br />
consequence ("Folgerungen") a theoretical law ("die Herleitung der aufgestellten<br />
Sätze": the principle of conservation of energy) 75; c) to compare this general<br />
principle <strong>with</strong> the empirical laws ("erfährungsmässigen Gesetzen") which connect<br />
the d) natural phenomena ("Naturerscheinungen") in various fields of physics 76.<br />
Helmholtz thus not only plans to offer, at variance <strong>with</strong> most of the other<br />
researchers involved <strong>with</strong> conservation problems, a specific functional<br />
formulation 77 of the quantities conserved <strong>and</strong> of their interrelations, but also a<br />
derivation of this "principle" from more general physical assumptions. This is an<br />
implicit assertion of the possibility of alternative versions of the principle.<br />
But the great theoretical innovation is that empirical laws are supposed to<br />
be compared no longer only <strong>with</strong> natural phenomena, but also <strong>with</strong> a general<br />
principle. It is not difficult to underst<strong>and</strong> Magnus' <strong>and</strong> Poggendorff's perplexities<br />
in the evaluation of the essay: the young physiologist <strong>with</strong>out presenting new<br />
experimental results adds two levels (a, b) to the st<strong>and</strong>ard practice of<br />
(experimental) physicists - that of formulating empirical laws (c) which would fit<br />
natural phenomena (d).<br />
One of the first conscious criteria of demarcation between theoretical <strong>and</strong><br />
empirical science can now be drawn 78: while the experimental scientist is looking<br />
for empirical generalisations that fit experimental data (e.g.: the refraction <strong>and</strong><br />
reflection laws), the theoretical scientist looks for the agreement of the principle<br />
of conservation <strong>with</strong> existing empirical laws (justificatory role of the principle)<br />
<strong>and</strong> for the theoretical discovery of new ones (heuristic role). Helmholtz here is<br />
explicitly setting out the task of theoretical research for the following decades:<br />
agreement <strong>with</strong> principles will become a condition which empirical laws have to<br />
satisfy, as important as the agreement <strong>with</strong> experimental data.<br />
73 Helmholtz Erhaltung P.1<br />
74 To be done in section 1<br />
75 In section 2<br />
76 In sections 3, 4, 5, 6.<br />
77 A general framework that allowed energy terms, while retaining some constant<br />
features, to acquire diferent expressions in different applications.<br />
78 Helmholtz Erhaltung P.2
The methodological skills of Helmholtz span on an even broader horizon:<br />
he believes that this already sophisticated approach has to be justified on<br />
philosophical grounds. Helmholtz was clever enough to realise that this would<br />
have been too much for contemporary physicists, <strong>and</strong>, as already recounted,<br />
initially dropped the Introduction, unfortunately <strong>with</strong>out achieving the goal of<br />
publication. The actual scope of the Einleitung, later restored <strong>with</strong> the addition of<br />
the premise, is to add an extra level to this already sophisticated pyramid of<br />
knowledge. In fact it was meant to show the meaning of the two initial<br />
assumptions for the final ("letzten") <strong>and</strong> true ("eigentlichen") goal ("Zweck") of<br />
physical sciences 79.<br />
Helmholtz believes that this means: a) to find the unknown causes of the<br />
phenomena, <strong>and</strong> b) to underst<strong>and</strong> them through the law of causality 80. While the<br />
unknown causes ("unbekannten Ursachen") are going to be explicitly identified<br />
<strong>with</strong> constant Newtonian forces 81, what is meant here by law of causality<br />
("Gesetze der Causalität") is more difficult to explain. My interpretation is that<br />
here Helmholtz refers to the theoretical search of the "empirical" link between<br />
natural phenomena <strong>and</strong> particularly to the causal link he is about to establish in<br />
chapter 2 between living <strong>and</strong> tension forces. Thus in my view the law of causality<br />
is taken here in its "regulative-empirical" sense 82, that is as a theoretical relation<br />
between "empirical" terms.<br />
But Helmholtz rapidly shifts to a different meaning of causality, a<br />
"transcendental" one, that is, causality seen as a precondition for the possibility<br />
of scientific knowledge: the scientist must assume that nature be intelligible, that<br />
"every transformation in nature must have a sufficient cause ( jede Veränderung<br />
in der Natur eine zureichende Ursache haben müsse)" 83. A natural process is<br />
79 Helmholtz Erhaltung Pp.1-2.<br />
80 Helmholtz in 1882 added here a note on causality <strong>and</strong> Kantianism; see below.<br />
81 Interesting remarks on <strong>Helmholtz's</strong> relationship between law <strong>and</strong> force as an<br />
antifunctionalist tool are in Lenoir:Strategy of Life at p.232; but Helmholtz did not assert "the<br />
conservation <strong>and</strong> interconvertibility of all central forces" P.231.<br />
82 For the distinction between "regulative-empirical" <strong>and</strong> "transcendental" causality in<br />
Kant see: Buchdahl, Gerd. "Reduction-Realization: A Key to the Structure of Kant's<br />
Thought". In Philosophical Topics XII Vol.2 1981-2: 39-98; specially Pp.83-4; Peter<br />
Heimann: "Helmholtz <strong>and</strong> Kant".Pp.221-3; in n.54 Heimann refers to (older works of)<br />
G.Buchdahl.<br />
83 Helmholtz Erhaltung P.2
intelligible if it can be referred to final causes. In fact final causes act following a<br />
constant law <strong>and</strong> thus, if the external conditions are the same (ceteris paribus),<br />
they produce the same effect.<br />
Helmholtz, probably in relation to the vital force debate, asserts that it<br />
might be possible that not all natural processes are actually intelligible 84. Some<br />
phenomena might belong to a realm of spontaneity <strong>and</strong> freedom, this cannot be<br />
decided conclusively, but the scientist must, all the same, assume the<br />
intelligibility of nature as the departure point for his investigations. Here, in my<br />
view, comes into play the second basic physical assumption to be justified, the<br />
impossibility of perpetual motion. The impossibility of perpetually providing<br />
work <strong>with</strong>out a corresponding compensation is in fact a limit to the spontaneity<br />
<strong>and</strong> freedom of nature <strong>and</strong> a physical version of the principle of sufficient cause.<br />
Often the Kantian character of the Einleitung has been stressed 85.<br />
However it has to be remarked that different parts of Kant's work play different<br />
roles here. Up to now Helmholtz has dealt <strong>with</strong> i) the regulative principle of<br />
empirical causality, ii) the transcendental principle of causality as the one<br />
granting the possibility of scientific knowledge <strong>and</strong> the lawlikeness of nature.<br />
Instead, <strong>Helmholtz's</strong> preoccupation in the following pages is iii) the conceptual<br />
explanation of a specific physical model, tending to show the possibility of<br />
Newtonian forces <strong>and</strong> not, at this stage, their inductive validity 86, The specific<br />
model is the Newtonian one, based on central forces depending only on distance.<br />
Through a detailed conceptual explanation based on the mechanistic categories of<br />
matter <strong>and</strong> force, Helmholtz tries to show that Newtonian forces can be<br />
considered the ultimate causes of natural phenomena <strong>and</strong> follows the method of<br />
Kant's Metaphysical Foundations of Natural Science . <strong>Helmholtz's</strong> famous<br />
84 Helmholtz Erhaltung P.2<br />
85 Helmholtz himself in 1882: W.A. 1 "Erhaltung" P.68; Ellington, J.W. "Kant". In<br />
C.C.Gillespie (ed.) DSB VII New York: Scribner, 1973. Pp.224-35; P.234; Elkana, Yehuda.<br />
"<strong>Helmholtz's</strong> 'Kraft': An Illustration of Concepts in Flux". In HSPS 2 (1970): 263-98;<br />
P.Heimann: "Helmh <strong>and</strong> Kant"; Wise in: Wise, Norton. "German Concepts of Force, <strong>Energy</strong><br />
<strong>and</strong> the Electromagnetic Ether: 1845-1880" In Conceptions of Ether. G.N. Cantor <strong>and</strong><br />
M.J.S.Hodge eds. Cambridge : Cambridge U.P., 1981. Pp.269-307, agrees <strong>with</strong> the influence<br />
of Kant's Metaphysical Foundations on Helmholtz, stressing the role of intensity <strong>and</strong> capacity<br />
factors; Fullinwider, S.P. "Hermann Von Helmholtz: The Problem of Kantian Influence". In<br />
Stud. Hist. Phil. Sci. 21 (1990): 41-55.<br />
86 see Heimann: "Helmholtz <strong>and</strong> Kant" P.229.
enunciation of the mechanical world 87 view is based on the assumption that both<br />
matter <strong>and</strong> force are abstractions <strong>and</strong> that the first cannot be considered more<br />
"real" than the second. Here he makes explicit that the problem of finding<br />
unchanging basic causes can be interpreted as the problem of finding constant<br />
forces. Causes <strong>and</strong> forces can be identified. A characteristic of the definition of<br />
force is that it is constant in time; bodies <strong>with</strong> constant forces only allow spatial<br />
movements <strong>and</strong> if the forces of extended bodies are decomposed into forces<br />
acting between material points, the intensity of forces depends only on the<br />
distances. This in <strong>Helmholtz's</strong> view is a direct consequence of the principle of<br />
sufficient reason ("Satz von zureichenden Grunde") 88. Thus if all natural<br />
phenomena could be reduced, through a general application of the principle of<br />
conservation, to the effects of attractive or repulsive forces whose intensities<br />
depend only on distance, then "empirical" causality would match "transcendental"<br />
causality <strong>and</strong> the task of physical science would be satisfied: an "intelligible"<br />
nature would be "understood".<br />
<strong>Helmholtz's</strong> conceptual model of forces was to be of special relevance in<br />
the whole energy debate. It was by no means universally accepted. On the<br />
contrary, Weber's electrodynamic law of 1846, based on the alternative<br />
assumption of forces depending on distances, velocities <strong>and</strong> accelerations, was<br />
gaining universal recognition. This might explain the particular care given in the<br />
Introduction to the conceptual explanation of the model. No confusion is made<br />
here between Kraft as energy <strong>and</strong> Kraft as Newtonian force: only the second<br />
meaning is applied 89.<br />
Helmholtz admitted that in the development of Mechanics the model of<br />
the forces had not yet been limited to the Newtonian one, but claimed that this<br />
was due to a lack of underst<strong>and</strong>ing of the foundations themselves of Mechanics.<br />
Helmholtz claimed moreover that some of the most important mechanical<br />
principles are valid only on the assumption of central forces depending on a<br />
distance. Among these the principle of living forces plays a special role<br />
(Helmholtz will "show" that the principle of virtual velocities can be deduced<br />
from the one of vis viva 90) <strong>and</strong> thus it can be considered the most general <strong>and</strong><br />
87 See Einstein <strong>and</strong> Infeld quoted n.164.<br />
88 Helmholtz Erhaltung P.5.<br />
89 Heimann, contra Elkana, shows that the meaning of "Kraft" is always unequivocal<br />
in the context: "Helmholtz <strong>and</strong> Kant" P. 207 n.10 <strong>and</strong> P.209 n.14.<br />
90 Helmholtz Erhaltung Pp.6 <strong>and</strong> 17-8.
elevant consequence of the deductions made (the great role that Helmholtz gives<br />
to the principle of vis viva will appear evident in the first chapter of the<br />
Erhaltung ).<br />
Helmholtz in 1882 added some appendices to the reprint of the<br />
Erhaltung; two of which are relevant for the present discussion of the<br />
Introduction. In the first 91 he asserted he is now (1881-2) less Kantian than in<br />
1847. But he also reasserted that the causality principle is nothing else than the<br />
presupposition of the lawlikeness of natural phenomena <strong>and</strong> reasserted the<br />
identification of cause, force <strong>and</strong> law. In my view, after the controversies <strong>with</strong><br />
the metaphysicians <strong>and</strong> his stress on the empirical aspects of geometry,<br />
Helmholtz wanted to detach himself from a rigid interpretation of the validity<br />
<strong>and</strong> reality of Kant's categories, while reasserting their use, in particular causality,<br />
as presuppositions. That is, <strong>with</strong> the terminology adopted here: detach himself<br />
from the interpretation of Newtonian forces as final causes but still adhere to the<br />
regulative <strong>and</strong> transcendental use of causality 92. Also the other Appendix show<br />
that, after the electrodynamic debate of the seventies, the Newtonian force model<br />
was shaken. I rather believe that Helmholtz wanted to detach himself from the<br />
Kantian model of Newtonian forces <strong>and</strong> instead stress the Kantian belief in the<br />
transcendental causality as condition for the possibility of experience, a lasting<br />
methodological tool for Helmholtz till the "Introduction to Theoretical lectures"<br />
of 1894.<br />
In the second Appendix 93 Helmholtz again tackled the problem raised by<br />
his model of central forces <strong>and</strong> reasserted, against criticisms, some aspects of his<br />
conceptual explanation: the discussions of chapter 1 <strong>and</strong> 2 of the Erhaltung can<br />
still be considered "partially" valid only if it is accepted that forces can be<br />
decomposed into point forces, <strong>and</strong> that the principle of superposition holds. But<br />
this last assumption has to be explicitly admitted <strong>and</strong> cannot be considered any<br />
longer a necessary consequence of the intelligibility of nature. Finally he<br />
91 Helmholtz WA1 P.68.<br />
92 Kahl Selected P. 49, Lindsay Applications P.27, Galaty "German Reductionism",<br />
<strong>and</strong> Fullinwider "Influence" P.53 wrongly interpret "stärker..., als ich jetzt" as "strongly, as I<br />
still" instead of "more strongly, than I now" as more correctly Heimann does: "Helmholtz <strong>and</strong><br />
Kant" P.219. But, while agreeing <strong>with</strong> Heimann that despite qualifications the appendix still<br />
reveals a Kantian framing, I do not believe that it was meant to "provide a stronger<br />
justificational foundation for the central forces principles". Ibid p.220.<br />
93 Helmholtz WA1 Pp.68-70.
mentioned the electrodynamic theories of Weber <strong>and</strong> Clausius that accepted<br />
forces depending not only on distances but also on velocities <strong>and</strong> accelerations.<br />
These theories are criticised: they contradict the principle of action <strong>and</strong> reaction<br />
<strong>and</strong> <strong>Helmholtz's</strong> formulation of energy conservation. Moreover, in the case of<br />
Clausius' law, the role played by empty space, if accepted, would imply the<br />
ab<strong>and</strong>onment of the goal of intelligibility: accepting something that acts but on<br />
which we cannot act would mean "a complete renunciation of the hope of<br />
solving completely the tasks of natural sciences" 94. Obviously a theoretical<br />
criticism.<br />
Thus, in 1882, Helmholtz showed he still adhered to the regulative <strong>and</strong><br />
transcendental use of causality but had problems <strong>with</strong> the conceptual model so<br />
carefully outlined in 1847. Indeed the model, while being essential for his specific<br />
formulation of energy conservation had been <strong>and</strong> still was the main target of<br />
criticisms.<br />
2 The two roots of the vis viva principle <strong>and</strong> their supposed equivalence<br />
In the first chapter Helmholtz tried to demonstrate the equivalence of his<br />
two basic assumptions (central forces <strong>and</strong> impossibility of perpetual motion)<br />
through an analysis of the vis viva principle. His argument starts <strong>with</strong> the<br />
enunciation of the principle of impossibility of perpetual motion:<br />
"We will start moving from the assumption that it is impossible, through<br />
any combination of natural bodies, to produce continually motive force from<br />
nothing" 95.<br />
Helmholtz asserted that Carnot <strong>and</strong> Clapeyron 96 had theoretically<br />
deduced a number of laws from this principle <strong>and</strong> that his own aim is to introduce<br />
the principle into every branch of physics "in the same way", to show both its<br />
justificatory (that is, its applicability to all the cases where the laws of the<br />
phenomena have been already determined) <strong>and</strong> its heuristic role (in offering a<br />
guiding thread to the experiments).<br />
94 Helmholtz WA 1 P.70.<br />
95 Helmholtz Erhaltung P.7.<br />
96 Already quoted in the "Bericht".
Needless to say the plan was to be carefully carried out, but not <strong>with</strong>out<br />
the introduction of subtle innovations of extraordinary relevance, which are not<br />
limited to overcoming the caloric model for heat still adopted by Carnot <strong>and</strong><br />
Clapeyron. The assertion "in the same way" refers to the methodology, but by<br />
no means indicates that the same expression of the principle was to be applied.<br />
Helmholtz in fact reformulates the principle of impossibility of perpetual motion,<br />
utilising the term "work" ("Arbeit"), <strong>and</strong> mechanical terms as "force" <strong>and</strong><br />
"velocity":<br />
"...the quantity of work obtained when a system of bodies moves from<br />
one position to another under the action of specific forces must be the same as<br />
that needed to carry the system back to the original position, independent of the<br />
way, the trajectory or the velocity of the change" 97.<br />
Thus, the term "Arbeit" is now a function of the state (position) of the<br />
system, it is a total differential : in a closed path work cannot be created, but<br />
cannot be destroyed. This is a first great innovation. The new concept of work,<br />
being now a function of the positions, can be equated to another function of the<br />
position: the vis viva.<br />
In fact from Galileo's relation<br />
v=√2gh,<br />
where v is the final velocity acquired by a body of mass m in a fall from<br />
height h under the acceleration g, we derive that<br />
the work mgh equates the expression<br />
,<br />
which is a function of the position too. Here Helmholtz uses again the<br />
word "Arbeit" for work, <strong>and</strong> following explicitly the French engineers' definition<br />
of"travail", in equating work <strong>and</strong> vis viva gives priority to the concept of work. In<br />
fact he defines<br />
<strong>and</strong> not as the measure of vis viva; in this way it<br />
"becomes identical <strong>with</strong> the quantity of work" 98.<br />
97 Helmholtz: Erh .P.8<br />
98 Compare this awareness of the role of the work concept <strong>with</strong> the difficulties in<br />
chapter 5 to adopt a definition of "potential in itself" equivalent to work. An indication of<br />
<strong>Helmholtz's</strong> theoretical rather than mathematical approach. See below.
This is done in order<br />
"to establish a better agreement <strong>with</strong> the customary way today of<br />
measuring the intensity of forces". 99<br />
Given the equivalence of work <strong>and</strong> vis viva the "mathematical expression"<br />
of the principle of impossibility of perpetual motion is obtained, that is, the law<br />
of the conservation of vis viva:<br />
"When any number whatsoever of mobile point masses moves solely<br />
under the influence of forces, which they exercise on each other, or that are<br />
produced by fixed centres, then the sum of the living forces of all the point<br />
masses together is the same at every instant of time at which all the points are in<br />
the same relative positions respect to each other <strong>and</strong> towards possible fixed<br />
centres, whichever their trajectories <strong>and</strong> their velocities during the time<br />
interval" 100.<br />
It is relevant to stress the specific meaning of "conservation" utilised: here<br />
the quantity conserved (vis viva) is conserved at specific positions <strong>and</strong> not during<br />
the process, a definition that echoes Huygens' results for the compound pendulum<br />
<strong>and</strong> Lagrange's definition of vis viva conservation.<br />
Helmholtz moreover wanted to show that the principle holds only if the<br />
forces can be decomposed into forces of mass points that are central. From 101:<br />
,<br />
where q is the velocity of a mass point m moving under the forces<br />
exerted by a fixed system A <strong>and</strong> x,y,z are the Cartesian coordinates, <strong>and</strong> from<br />
where X,Y,Z are the component of the acting forces <strong>and</strong> dq=Xdt/m,<br />
Helmholtz derived incorrectly ( by equating the corresponding components of the<br />
second members):<br />
99 Helmholtz : Erh. P.9. Helmholtz did not rederive the definition by himself as<br />
instead suggested by Kuhn: Sim Disc p.88.<br />
100 Helmholtz : Erh. P.9; the formulation was to be criticised in 1854 by Clausius for<br />
whom the positions cannot be considered as "relative" but are actually referred to fixed<br />
centres.<br />
101 Helmholtz: ibid.P.11.
<strong>and</strong> from here that the direction <strong>and</strong> magnitude of the force must be<br />
function of the position of m <strong>and</strong> thus of its distance from the attracting point a.<br />
Helmholtz synthesized in this chapter many elements deriving from<br />
different traditions <strong>and</strong> introduced many novelties sometimes implicitly <strong>and</strong> not<br />
always <strong>with</strong> success.<br />
The first remark to be made concerns the formulation of the impossibility<br />
of perpetual motion: Helmholtz introduced specific mechanical concepts (work,<br />
velocity, force) that did not belong to the Carnot-Clapeyron expression. There is<br />
thus an attempt at framing the impossibility of perpetual motion in a mechanical<br />
world view 102: an (implicit) step in the methodological strategy tending to show<br />
that the two initial assumptions belong to the same conceptual scheme.<br />
My second remark deals <strong>with</strong> the relevant innovation introduced by<br />
Helmholtz, that is the interpretation of the term work ("Arbeit") as a total<br />
differential in the (new) expression of the impossibility of perpetual motion. Here<br />
Helmholtz unified two different traditions in mechanics, analytical mechanics<br />
<strong>and</strong> mechanical engineering 103, <strong>and</strong> an old philosophical principle, partially<br />
already recalled in the physiological discussions of the "Bericht": "nothing comes<br />
out of nothing <strong>and</strong> nothing is destroyed" 104.<br />
In the French tradition of mechanical engineering, which, as seen, was<br />
well known to Helmholtz, the term "travail" had received full importance: the<br />
principle of conservation of vis viva became the principle of transmission of<br />
work 105; but while accepting the impossibility of creating work the French<br />
engineers did not exclude that work could be lost 106. Being mostly concerned<br />
102 Helm's criticism on this point is very sharp: he summarised the mechanical<br />
hypothesis <strong>with</strong> the statement that "all the events can be explained in terms of forces that<br />
originate accelerations". He warned that <strong>with</strong> <strong>Helmholtz's</strong> reformulation of the principle of<br />
impossibility of perpetual motion <strong>and</strong> of the subsequent expression of the vis viva principle,<br />
whenever the last one is applied we will be "obliged to imagine forces <strong>and</strong> velocities involved<br />
in every experimental application, for instance in electrical or thermal phenomena". For Helm<br />
the two roots of <strong>Helmholtz's</strong> Erhaltung are basically different <strong>and</strong> the one based on<br />
Newtonian forces has to be rejected, together <strong>with</strong> the mechanical formulation of energy.<br />
Helm Energetik P.41.<br />
103 for a classification see: Grattan-Guinness "The varieties of mechanics"<br />
104 WA 1 P.6<br />
105 Rühlmann Maschinenlehre ; Haas Entwickl. Pp.73-83.<br />
106 Haas: ibidem P.81.
<strong>with</strong> impacts, for them work was not a total differential <strong>and</strong> the concept of<br />
potential, which was being developed, not only in France 107, in the tradition of<br />
analytical mechanics, was not generally admitted 108. On the other side in the<br />
analytical tradition, up to a certain date, the quantity that is now called potential<br />
was not meant to be work stored in the system at a certain position, but, despite<br />
formal equivalence, was understood only as a mathematical function of the<br />
positions from which the forces could be derived. Force by displacement in the<br />
direction of force was in this tradition a total differential but did not receive a<br />
physical interpretation. Helmholtz very subtly <strong>and</strong> skilfully here unified the two<br />
approaches, that is, the concept of work <strong>with</strong> the function of positions (but not<br />
<strong>with</strong>out problems in defining potential 109). Now work cannot be created <strong>and</strong><br />
cannot be destroyed, it is a state function (of the positions). Discussions in the<br />
1880's outlined clearly that <strong>with</strong> this new "representation" of the principle of<br />
impossibility of perpetual motion a relevant modification had been achieved 110.<br />
The relevance of the innovation just introduced was made explicit by<br />
Helmholtz himself in 1884 111; he asserted that there are two philosophical roots<br />
of the principle of conservation: the "ex nihilo nil fieri" <strong>and</strong> the "nil fieri ad<br />
nihilum". The first is connected <strong>with</strong> the impossibility of creating work ( <strong>and</strong> thus<br />
<strong>with</strong> the impossibility of perpetual motion) <strong>and</strong> the second <strong>with</strong> the impossibility<br />
of destroying it. Helmholtz in 1884, looking back, asserted that a big difficulty<br />
he had to overcome in the formulation of energy conservation was the acceptance<br />
of the "ad nihilum", while the first root was part of shared knowledge 112. The "ex<br />
nihilo" is based on an inductive assumption, already made by a qualified minority<br />
nn.71-4.<br />
107 see for instance the contributions of Green, Hamilton, Gauss, F.Neumann: above<br />
108 Grattan-Guinness, Ivor. "Work for the Workers: Advances in Engineering<br />
Mechanics <strong>and</strong> Instruction in France, 1800-1830". In Annals of Science 41 (1984):1-33. P.<br />
32.<br />
109 See chapter 5 of the Erhaltung<br />
110 Planck Princip P. 37.<br />
111 In an appendix to the reprint of the famous talk on the Interaction of Natural<br />
Forces: Helmholtz, Hermann. "Robert Mayer's Priorität". In Vorträge und Reden. 2 vols.<br />
Braunschweig: Vieweg, 1884.<br />
112 Both had been already explicitly recalled by Mayer in 1842.
since Leonardo 113; the "ad nihilum" was more difficult to accept, given the few<br />
experiences aimed at the destruction rather than the production of work (<strong>and</strong> this<br />
can explain the above mentioned approach of the French engineers) 114.<br />
That Helmholtz apart from being aware of the French theoretical<br />
engineering tradition was also fully aware of the tradition of analytical<br />
mechanics can also be seen 115 in the similarities between his definition of the<br />
principle of vis viva conservation <strong>and</strong> the one of Lagrange. Moreover the<br />
"derivation" itself of this principle through Galileo's free fall law echoes the<br />
historical account of the development of the principle of vis viva conservation<br />
given in Lagrange's Mechanique Analitique<br />
The latter's account 116 can be summarised in the following steps:<br />
Descartes had defined work as force (weight) by displacement; from the<br />
impossibility of perpetual motion Leibniz showed that vis viva ( ) is the "true"<br />
measure of the capacity of doing work of a body in motion; Huygens through his<br />
compound pendulum obtained the result that the sum of the vis viva of a system<br />
of bodies is a function of the positions <strong>and</strong> is independent of the constraints of the<br />
system <strong>and</strong> of the way in which the system of bodies reverts back to those<br />
positions ( here of course the velocity v is the final velocity that a body in a given<br />
position would acquire when "falling" from that position to a second position<br />
assumed as reference). D.Bernoulli, utilising Galileo's free fall relation, stated<br />
that the vis viva is equal to a product of force times a distance. D'Alembert <strong>and</strong><br />
Euler clarified the analytical relations between vis viva <strong>and</strong> work discarding<br />
Leibniz's philosophical ideas on causality <strong>and</strong> utilising Newton's definition of<br />
113 See the interesting pages of Leonardo, Cardano, Stevin <strong>and</strong> Galileo in: Lindsay,<br />
Robert Bruce. <strong>Energy</strong>, Historical Development of the Concept. Stroudsburg: Dowden<br />
Hutchinson <strong>and</strong> Ross, 1975.<br />
114 Planck in 1887 gave great relevance to the two roots (chapt.1) <strong>and</strong> showed that<br />
the principle of conservation of energy cannot be deduced from the impossibility of perpetual<br />
motion only but that also the "ad nihilum" is needed (chapt.2). Planck Princip P.3, Pp.138-9.<br />
Helm in 1898 still underlined the need for the two roots <strong>and</strong> quoted Planck's demonstration:<br />
Energetik P.51.<br />
115 apart from always dubious agiographical claims: Koenigsbeger: H v H P.30 (for<br />
<strong>Helmholtz's</strong> knowledge of Jacobi's Funct.Ellipticarum ) <strong>and</strong> P.43 (knowledge of D.Bernoulli<br />
<strong>and</strong> D'Alembert; also: Jacobi recognised links between the Erhaltung <strong>and</strong> the analytical<br />
tradition).<br />
116 Lagrange, J.L. Mechanique Analitique , Paris: Desaint, 1788.
force. Lagrange himself in the Mechanique Analitique from the principle of<br />
virtual work deduced the vis viva principle outlining that the name conservation<br />
of vis viva is given because the sum of the vis viva expressions of the bodies of<br />
the system is the same when the bodies revert back to their original positions,<br />
independently of the trajectories followed. Thus, for Lagrange, conservation of<br />
vis viva is, after Huygens, conservation at specific positions 117 <strong>and</strong> not<br />
conservation during a process. In Lagrange's formulation 118:<br />
the stress is on the vis viva term: the factor 2 is <strong>with</strong> the second member;<br />
the function of the positions (the potential term) <strong>and</strong> the constant F are not<br />
given special relevance <strong>and</strong> do not acquire physical meaning, despite the fact<br />
that, from a contemporary point of view, the two terms on the right side of the<br />
equation are easily identified <strong>with</strong> total energy <strong>and</strong> potential energy of the<br />
system 119.<br />
My third remark outlines the criticisms to <strong>Helmholtz's</strong> "demonstration"of<br />
the equivalence of the two initial assumptions. This "wrong" demonstration plays<br />
a vital role in <strong>Helmholtz's</strong> research program: the generalisation of the principle of<br />
conservation of vis viva into <strong>Helmholtz's</strong> principle of conservation of "force", that<br />
is, into a principle where the kinetic <strong>and</strong> positional terms can be sharply split, is<br />
only possible if the forces depend solely on distances. While this applies to<br />
Newton's, Coulomb's <strong>and</strong> Ampere's forces, it does not apply to Weber's<br />
electrodynamic ones, not a minor problem for Helmholtz. It being impossible at<br />
the time to oppose Weber's law on empirical grounds 120, it was important for<br />
Helmholtz to demonstrate that empirical laws which admitted forces different<br />
from the central ones violated the conservation of vis viva <strong>and</strong> the impossibility<br />
of perpetual motion.<br />
117 This particular meaning of conservation surprises some commentators: Lindsay<br />
(unhistorically) wonders why the name of conservation of vis viva is given if vis viva is<br />
conserved only in the specific case of constant potential energy: Hist Devel Pp. 167-9.<br />
118 Lagrange Mech An P.208.<br />
119 The same can be said of other expressions of vis viva conservation typical of<br />
analytical mechanics: P.S.Laplace, Pierre Simon.Traité de la Mécanique Celeste . 5 vols.<br />
Paris, 1799-1825. Vol I, P. 52; Hamilton "On a General Method in Dynamics" Phil Tr Roy<br />
Soc Pt II 1834: 247-57. P.250; in: Lindsay Hist P.264.<br />
120 Helmholtz will try that in the seventies.
But unfortunately for the whole strategy this was an impossible task.<br />
<strong>Helmholtz's</strong> demonstration was based on a wrong assumption: even if the<br />
components of the vis viva depend on the positions only, the same does not<br />
necessarily follow for the force components. In fact it was possible to show that<br />
forces depending on velocities <strong>and</strong> accelerations, like Weber's, do not violate the<br />
conservation of vis viva or the impossibility of perpetual motion: Weber in 1848<br />
was to show that his own force admitted a potential, even if a kinetic one 121. But<br />
all the same <strong>Helmholtz's</strong> approach had for twenty years an incredible success: in<br />
the British literature the point that Weber's force law denied conservation of vis<br />
viva <strong>and</strong> energy conservation was maintained by Maxwell 122 in 1865, Tait <strong>and</strong><br />
Thomson in 1867 123 <strong>and</strong> Tait in 1868 124, <strong>and</strong> finally refused by Maxwell in 1873<br />
only after <strong>Helmholtz's</strong> own retreat in 1870. Helmholtz himself acknowledged in<br />
1882 125 that a flaw in the 1847 demonstration had been shown by Lipschitz (he<br />
still did not mention Clausius' criticisms 126). Nevertheless he had to agree that he<br />
121 The first to explicitly criticise the limitations of the validity of the principle of vis<br />
viva to central forces was Clausius in 1852 <strong>and</strong> 1854.<br />
122 Maxwell, James Clerk. "A Dynamical Theory of the Electromagnetic Field". Repr.<br />
in: Scientific papers. 2 Vols. Cambridge: Cambridge U.P., 1890. Vol.1 Pp.526-597; at<br />
pp.526-7.<br />
123 Thomson,W. <strong>and</strong> Tait, P.G. Treatise on Natural Philosophy. , Clarendon Press,<br />
Oxford, 1867. Par. 385. P.311-2.<br />
124 Tait,P.G. Sketch of Thermodynamics., Edmonston <strong>and</strong> Douglas, Edinburgh, 1868.<br />
P.76. See also C.Neumann's defense of Weber : Neumann, Carl. Die Gesetze von Ampére und<br />
Weber. Leipzig: Teubner, 1877. Pp.322-4; for a discussion see my <strong>Energy</strong> <strong>and</strong> Electr Pp.122-<br />
3 <strong>and</strong>133-6.<br />
125 Helmholtz WA 1 P.70.<br />
126 Clausius, Rudolf. "Ueber das mechanische Aequivalent einer elektrischen<br />
Entladung und die dabei stattfindende Erwärmung des Leitungsdrahtes" In Pogg<br />
Ann 86 (1852): 337-375<br />
Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber<br />
die Erhaltung der Kraft." In Pogg Ann 89 (1853): 568-579.<br />
Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber<br />
die Erhaltung der Kraft", zweite Notiz." In Pogg Ann 91 (1854): 601-604.
had not been able to "demonstrate" that central Newtonian forces had a<br />
privileged status 127.<br />
<strong>Helmholtz's</strong> foundations of the Erhaltung are wide ranging but hide a not<br />
insignificant weakness: the equivalence between the two conceptual roots<br />
(impossibility of perpetual motion <strong>and</strong> the resulting conservation of vis viva on<br />
one side, <strong>and</strong> the forces depending only on the distance on the other) had not<br />
been <strong>and</strong> could not be demonstrated. This was to become explicit very soon.<br />
Based thus on unsecure deductive grounds the validity of <strong>Helmholtz's</strong> approach<br />
had to rely on the inductive side: on its "empirical" success; that is on its<br />
capability of reassessing already existing knowledge (justificatory power) <strong>and</strong> on<br />
the capability of disclosing the interpretation of new phenomena (heuristic<br />
power).<br />
3 The duck, the rabbit <strong>and</strong> the principle of energy conservation.<br />
Helmholtz was now ready for a great generalisation: in the second chapter<br />
the principle of conservation of vis viva will become the principle of conservation<br />
of "force".<br />
127 As already recalled Helmholtz still in 1882 tried to save his original approach <strong>with</strong><br />
the introduction of a new hypothesis: the forces that do not obey a central Newtonian force<br />
law cannot fulfill the action <strong>and</strong> reaction principle. Helm in 1898 restated Clausius' criticisms<br />
but from a different perspective: Clausius shared the mechanical point of view <strong>and</strong> attacked the<br />
privilege given to the Newtonian forces, Helm instead criticised the mechanical world view<br />
<strong>and</strong> the supposed equivalence of the two roots of Helmholtz: the flaw in the demonstration<br />
shows, according to Helm, the independence of the principle of perpetual motion from the<br />
model of Newtonian central forces. A relevant remark is shared by Planck <strong>and</strong> Helm:<br />
Helmholtz, together <strong>with</strong> the impossibility of perpetual motion, made implicit use of another<br />
principle to avoid the problems created for his own approach by the irreversibility of many<br />
transformations in nature: the principle that a state, derived from another, can always be<br />
reconducted to this one even if through a different pattern. Helm: Energetik Pp.36-7.
The principle of conservation of vis viva for a point mass m, moving <strong>with</strong><br />
velocity q, along the path r, under the action of a central force , can be written<br />
as:<br />
or if Q <strong>and</strong> q are the velocities at the distances R <strong>and</strong> r:<br />
128<br />
This is formally identical <strong>with</strong> the well known theorem of vis viva-work.<br />
The first term is in fact the well known variation of the vis viva <strong>and</strong> the second<br />
term has the dimension of work (force by elementary displacement in the<br />
direction of the force integrated along a line). In the first chapter Helmholtz often<br />
utilized the word "Arbeit" so we would expect it mentioned again here. Instead a<br />
bold reinterpretation of the equation takes place, the second member of the<br />
equation is in fact not defined as "Arbeit" but as:<br />
"the sum of tension forces ( Spannkräfte) between the distances R <strong>and</strong><br />
r". 129<br />
Helmholtz made an effort at clarifying the innovation: the tension force<br />
was meant to be in explicit conceptual duality <strong>with</strong> the living force ("in contrast<br />
to that which Mechanics calls living forces" 130), a "force" that tends to move the<br />
point m, until movement actually takes place. A geometrical interpretation of the<br />
concept is given, it represents: "the set of all the intensities of the force acting in<br />
the distances between R <strong>and</strong> r". In fact if the intensities of correspond to<br />
ordinates perpendicular to the line of abscissae connecting the point m <strong>and</strong> the<br />
centre of force a, the integral represent an area given by the "sum of the infinite<br />
abscissae (read: ordinates) lying on it".<br />
The partly unsuccessful effort (the integral is not the sum of the abscissae<br />
but of infinitesimal surfaces) to give a geometrical interpretation of the concept of<br />
Spannkraft, stresses the fact that the tension forces dr are also very different<br />
from the Newtonian forces : dimensionally they are represented by the product<br />
of a force by a displacement; they exist when the material point is not in motion,<br />
tend to put it in motion <strong>and</strong> are "consumed" by the acquired motion (compare<br />
<strong>with</strong> the constant relation force-matter described as conceptual model for the<br />
128 Helmholtz Erhaltung p.13<br />
129 Helmholtz Erhaltung p.14<br />
130 Ibid
forces in the Introduction); while they are function of a distance (two positions)<br />
they acquire a proper meaning only when summed over a definite interval.<br />
Helmholtz introduced relevant new theoretical concepts in an old<br />
equation. Now both the first <strong>and</strong> second member have a physical theoretical<br />
meaning, connected by an equality that does hold during a process: a variation of<br />
one member equates the variation of the second member. The sum of the two<br />
members is physically interpreted as a third theoretical concept, later to be called<br />
energy:<br />
" 'In all cases in which free material points move under the influence of<br />
attractive <strong>and</strong> repulsive forces: forces, whose intensity depends only on the<br />
distance, the loss in quantity of tension force is always equal to the gain in living<br />
force, <strong>and</strong> the gain in the first is always equal to the loss of the second. Therefore<br />
the sum of the existing tension <strong>and</strong> living forces is always constant.' In this quite<br />
general form we can define our law as the principle of conservation of force." 131<br />
In the case of Huygens' conservation of vis viva, the meaning of<br />
conservation itself was different: conservation meant that the vis viva of a system<br />
reacquires the same value when the system reacquires the same positions,<br />
independently of the trajectories to revert back to those positions; of course<br />
velocity, <strong>and</strong> thus vis viva, changes during motion.<br />
<strong>Helmholtz's</strong> conservation instead introduced a new meaning of<br />
conservation: Kraft (energy) is conserved during motion <strong>and</strong> a variation of vis<br />
viva corresponds to an opposite variation of tension force.<br />
Finally Helmholtz presented the deduction of the principle of virtual<br />
velocities from the conservation of force 132 mentioned at the end of the<br />
Introduction: an increase of vis viva can result only from the consumption of a<br />
quantity of tension force. Thus a system at rest, if there is no consumption of<br />
tension forces for every possible direction of motion in the first instant, stays at<br />
rest.<br />
Helmholtz summarized the results so far obtained: a) the principle of<br />
conservation of force implies that the maximum quantity of work that can be<br />
obtained from a system is a determined, finite quantity if the acting forces do not<br />
depend on time <strong>and</strong> velocities ; b) if they do or if the forces act in directions<br />
131 Helmholtz Erhaltung p.17<br />
132 Clausius was to invert the priority <strong>and</strong> deduce the conservation of vis viva from<br />
the virtual velocities principle, more in line <strong>with</strong> the analytical tradition, in his 1852 paper <strong>and</strong><br />
in the various editions of his Potential <strong>and</strong> potential function.
different from the one joining the active material points "force" could be gained<br />
or lost ad infinitum; c) if forces were different from central, a system of bodies at<br />
rest could be set in motion by the effect of its own internal forces 133. The<br />
hypothesis of central forces depending only on distances is thus basic to<br />
<strong>Helmholtz's</strong> view, but these three results are not <strong>with</strong>out problems as seen in the<br />
previous subsection.<br />
In my view <strong>Helmholtz's</strong> summary of his second chapter does not do<br />
justice to the results he had obtained. A real shift in meaning occurred in fact: the<br />
well-known old equation written at the beginning of the chapter had acquired a<br />
new interpretation; we already knew that the stress could be <strong>with</strong> the tradition of<br />
analytical mechanics on the first term (the duck : conservation of vis viva), <strong>with</strong><br />
mechanical engineering on the second term (the rabbit: transmission of work);<br />
now <strong>with</strong> Helmholtz the stress was on the equivalence between the two.<br />
The introduction of the term Spannkraft reveals a real meaning shift: <strong>with</strong><br />
the tension force we are very far from the concept of work <strong>and</strong> very close to the<br />
idea of potential energy. It is not in fact work done but work that can be done,<br />
capacity to do work. Work acquires now the role of a unit of measurement for a<br />
new theoretical concept. Planck outlined the great importance of the step<br />
undertaken:<br />
"However insignificant this interpretation might seem at first glance, the<br />
perspective that it opens on all fields of physics is nevertheless extraordinarily<br />
wide, because now the generalisation to every natural phenomenon is evident." 134<br />
In 1887, <strong>with</strong> prophetic insight, Planck declared 135 also that <strong>with</strong><br />
<strong>Helmholtz's</strong> formulation the principle of conservation of "force" became similar<br />
to the one of conservation of matter: "force" as matter cannot be increased or<br />
diminished, but can appear in different forms. The two basic forms of "force", vis<br />
viva <strong>and</strong> tension force, can appear in many ways: vis viva as motion, light, heat;<br />
tension force as elevation of a weight, elastic or electric potential, chemical<br />
difference 136, <strong>and</strong> so on.<br />
"But the sum of all these reserves of force (so to say accumulated in<br />
different stores) is invariably the same <strong>and</strong> all natural processes only consist in<br />
133 Helmholtz Erhaltung Pp.19-20<br />
134 Planck Princip P.37.<br />
135 Planck Princip P.37<br />
136Probably in the sense of different energy levels connected <strong>with</strong> chemical bonds.
the simple passage from one to the other" 137. This was "another step forward<br />
towards simplifying the underst<strong>and</strong>ing of all natural phenomena" 138.<br />
It is indeed surprising that such a great innovation is seen as a failure by modern<br />
commentators 139 <strong>and</strong> discarded; I believe instead that an analysis of its origin<br />
outlines some specific features of a theoretical <strong>and</strong> a mathematical approach to<br />
physics. This can be shown relating <strong>Helmholtz's</strong> approach <strong>with</strong> Leibniz's one <strong>and</strong><br />
comparing it <strong>with</strong> Clausius's one.<br />
In my opinion, to achieve his result, Helmholtz made use of a Leibnizian<br />
inheritance 140. First the duality lebendig Kraft-Spannkraft strongly resembles the<br />
older one between vis viva-vis mortua; the terms themselves are very similar:<br />
while the vis viva has almost the same expression, the meaning of Spannkraft is<br />
very close to the older Leibnizian counterpart:<br />
"the ..... vis viva, produced by an infinite number of applications of vis<br />
mortua " 141<br />
But between <strong>Helmholtz's</strong> <strong>and</strong> Leibniz's ideas of positional "energy" there<br />
are two main differences that explain the success of the more recent formulation:<br />
a) Helmholtz provided a formal quantitative expression; b) the Newtonian forces<br />
are part of it.<br />
Helmholtz, making proper use of the Newtonian concept of force, <strong>and</strong><br />
accepting the full inheritance of Newtonian mechanics, provided a formal<br />
expression for the second term of the duality that was missing in Leibniz.<br />
But other Leibnizian elements can be outlined: the equality of the two<br />
terms in the mathematical expression is no longer the indication of an analytical<br />
identity. Being two independent physical concepts, now the equality has the<br />
meaning of a causal relation: the variation of one member implies the variation of<br />
the other. The equality holds at every instant during a process. This is a<br />
Leibnizian concept of conservation:<br />
137 Planck Princp P.37<br />
138 See Planck Princip P.37<br />
139 Kuhn thinks that Helmholtz "fails to recognize" the integral as "Arbeit", see above<br />
n.19; Lindsay too is surprised for the lack of the term "Arbeit" <strong>and</strong> stresses the mistake of<br />
considering an integral as the sum of lines. Lindasy Applic. of <strong>Energy</strong> P. 16.<br />
140 A Leibnizian influence on Mayer has been often remarked, on Helmholtz it has<br />
been only briefly pointed out by Planck Princip P.35, Koenigsberger H v H P.49; Elkana<br />
Disc Chapt 1 n.31.<br />
141 Leibniz : Specimen Dynamicum (1695) quoted in Lindsay Hist P.122.
"The equality that exists between effect <strong>and</strong> efficient cause confirms what<br />
we have just said. In this equality consists the conservation of forces of bodies<br />
that are in motion" 142.<br />
"Empirical" causality then comes into play here: not causality as a<br />
condition of the possibility of natural laws, but a principle which establishes a<br />
specific link between different realms of phenomena. The "empirical" causality<br />
indicates a quantitative equivalence between phenomena that are qualitatively<br />
different (static <strong>and</strong> dynamic).<br />
A static cause can generate, as an effect, the movement of a body. This<br />
movement, in its turn the cause, has the power ("motive force") to produce the<br />
effect of bringing the body back to its former position. The quantitative<br />
equivalence of cause <strong>and</strong> effect is supposed to hold true at every instant of the<br />
process: the interchangeability of the initial <strong>and</strong> final stage is only an<br />
exemplification of this principle. What is conserved during the process in<br />
Helmholtz as in Leibniz is this specific quantitative equivalence between<br />
qualitatively different phenomena.<br />
But how can we actually measure two qualitatively different phenomena<br />
to establish a quantitative equivalence? A common unit of measurement is<br />
needed. Again a problem already faced by Leibniz, who asserted the relevance of<br />
(what was later defined as) work as a unit of measurement of all natural<br />
phenomena 143. Leibniz had recognised the impossibility of continually creating<br />
work <strong>with</strong>out a corresponding compensation, <strong>and</strong> also of destroying work, i.e. of<br />
both the "ex nihilo" <strong>and</strong> "ad nihilum", a necessary condition to guarantee the<br />
invariability of the chosen unit.<br />
142 Johann Bernoulli Discours sur les lois de la communication du mouvement, Paris<br />
1724; chapt.V, par. 8; quoted in Lindsay Hist P.125.<br />
143 Leibniz:"Brevis Demonstratio erroris memorabilis Cartesii et aliorum circa legem<br />
naturalem, secundam quam volunt a Deo e<strong>and</strong>em semper quantitatem motus conservari, qua et<br />
in re mechanica abutuntur" in Acta Eruditorum, Christofory Guntheri, Lipsiae 1686, Pp.161-3;<br />
repr. in Leibniz Mathematische Schriften, 7 volumes, C.I.Gerhard ed, Berlin: Asher <strong>and</strong> Halle:<br />
Schmidt, 1849-63; Vol 2, 1860, Pp.117-9; more generally see: Cassirer, Ernst. Leibniz' System<br />
in seinen wissenschaftlichen Grundlagen, Marburg: Elwert, 1902; Substanzbegriff und<br />
Funktionsbegriff, Berlin: B.Cassirer, 1923, Chapt 4, Sect 7, pp.226-48; Das<br />
Erkenntnisproblem in der Philosophie und Wissenschaft der Neueren Zeit , 2 vols. 2nd<br />
revised edition, B.Cassirer: Berlin, 1911; vol 2. P.165.
In <strong>Helmholtz's</strong> essay too, work is the common unit of quantitative<br />
measurement for different phenomena, phenomena connected by a causal<br />
principle, but this time mechanically interpreted under Newton's definition of<br />
force <strong>and</strong> laws of motion. The "ex nihilo" <strong>and</strong> "ad nihilum" are now both present:<br />
the quantitative aspects of one member of the equation have to be the same as of<br />
the other member, no more nor less. Work cannot be created, but neither<br />
destroyed <strong>and</strong> thus a great generality is acquired.<br />
<strong>Helmholtz's</strong> program is thus that all natural phenomena be measured by a<br />
common unit <strong>and</strong> be interpreted through only two forms of "energy": <strong>with</strong>out<br />
doubt a great theoretical unification resulting from the above mentioned meaning<br />
shift.<br />
A deeper underst<strong>and</strong>ing of <strong>Helmholtz's</strong> result can be reached, in my opinion,<br />
through a comparison <strong>with</strong> a less philosophically but more mathematically<br />
inclined approach: the one of Clausius. This will be done in the next chapter.<br />
Another interesting comparison is the one between Helmholtz <strong>and</strong> Mayer: It is<br />
surprising that Helmholtz, despite being acquainted <strong>with</strong> Liebig's papers was not<br />
aware of Mayer's 1842 contribution to the Annalen der Chemie und Pharmacie .<br />
Nowhere in the Erhaltung Helmholtz quoted Mayer <strong>and</strong> he later claimed not to<br />
have known in 1847 of his paper. Still it is relevant to point out some similarities<br />
<strong>and</strong> differences between the two approaches : the use of a Leibnizian causality<br />
principle was in common, but the mechanical conception of nature , the<br />
mechanical theory of heat, the central force hypothesis <strong>and</strong> the reduction of all<br />
the qualitatively different forms of "force" to two basic ones were specific of<br />
Helmholtz. Mayer disagreed <strong>with</strong> all these elements <strong>and</strong> his expression of<br />
conservation of energy was closer to a principle of equivalence or in other words<br />
to a correlation principle. On the other side Mayer had worked out a mechanical<br />
equivalent of heat, even if not through original experiments but definitely through<br />
original thinking, while, as I will show in subsection D, Helmholtz had not 144.<br />
144 On Mayer see: the Tyndall-Tait debate of 62-65; above n.3; <strong>Helmholtz's</strong> remarks<br />
in the 1882 appendix to the Erhaltung : WA 1 Pp.71-3, Helmholtz, Hermann. "Robert Mayer's<br />
Priorität". In Vorträge und Reden. 2 vols. Braunschweig: Vieweg, 1884; <strong>Helmholtz's</strong> appendix<br />
to " Das Denken in der Medecin". In Vorträge und Reden. 2 Vols. Braunschweig: Vieweg,<br />
1884; see also Planck Princip Pp.21-8; Helm Energetik Pp.16-28; Haas Entwickl Pp.61-2;<br />
more recently: Lindsay, Robert Bruce. Julius Robert Mayer, Prophet of <strong>Energy</strong>. Oxford <strong>and</strong><br />
New York: Pergamon Press,1973; Heimann, Peter. "Mayer's Concept of 'Force': the 'Axis' of a<br />
New Science of Physics." In HSPS 7 (1976) : 227-96.
In virtue of its limitations to Newtonian forces <strong>Helmholtz's</strong> "Erhaltung der Kraft"<br />
achieved in the second chapter, <strong>with</strong> the sharp distinction between kinetic <strong>and</strong><br />
positional terms, is the fulfilment of the far- reaching plan anticipated in the first<br />
lines of the "Einleitung" : the conservation law just formulated is the<br />
"consequence" (level B) of the two basic physical assumptions (level A:<br />
impossibility of perpetual motion <strong>and</strong> central forces).<br />
But so far we only have a theoretical framework that has to be applied, i.e. filled<br />
<strong>with</strong> the specific expressions of the vis viva <strong>and</strong> tension forces resulting from the<br />
interaction of the principle (level B) <strong>with</strong> the experimental laws (level C) in the<br />
various realms of natural phenomena (level D). This interaction between levels B<br />
<strong>and</strong> C will be carried forward in the next four sections.<br />
Nevertheless, <strong>Helmholtz's</strong> achievements in the Einleitung <strong>and</strong> in the first two<br />
chapters are impressive. In a limited number of pages a complete reappraisal of<br />
previous traditions in physics has been made, <strong>with</strong> an original synthesis of<br />
different <strong>and</strong> previously competing approaches (analytical mechanics, Newtonian<br />
mechanics, Leibnizian philosophy, mechanical engineering ). But the programme<br />
is only at its initial stage : the general framework is going to be applied in the<br />
following four sections to an extremely wide variety of natural phenomena <strong>and</strong><br />
empirical laws. It is in this way that the "empty" framework will show its<br />
justificatory <strong>and</strong> heuristic power, but not <strong>with</strong>out problems. In my view it is<br />
useful to analyse in the following chapters of the Erhaltung the difficulties<br />
Helmholtz had to face in specifying the "energy" equations in the different<br />
branches of natural sciences. This in order to assess the merits <strong>and</strong> demerits of<br />
his mechanical approach based on the central force assumption, <strong>with</strong> the resulting<br />
distinction between the two main forms of energy. An evaluation of <strong>Helmholtz's</strong><br />
results in the application of the theoretical framework to the empirical laws of<br />
natural phenomena is a kind of analysis well known at the turn of the century,<br />
but forgotten in more recent times.<br />
4 An easy start: Mechanics<br />
In the third chapter of the Erhaltung Helmholtz started to apply the principle to<br />
mechanical theorems, mostly recalling applications already known of vis viva<br />
conservation. That is: in this short <strong>and</strong> non- mathematical chapter Helmholtz did
not deal <strong>with</strong> specific applications of the concept of tension forces. He<br />
considered briefly: the motions caused by gravitation, both of celestial <strong>and</strong><br />
terrestrial bodies; the transmission of movements through incompressible solids<br />
<strong>and</strong> fluids, if vis viva is not lost through friction or inelastic collisions; the<br />
motions of perfectly elastic 145 solid <strong>and</strong> fluid bodies <strong>with</strong>out internal friction 146. It<br />
is to be noted here that, dealing <strong>with</strong> wave motions, Helmholtz asserted that<br />
"likely" light <strong>and</strong> radiant heat have to be considered, together <strong>with</strong> liquids <strong>and</strong><br />
sounds, thus showing an early acceptance of Melloni's theory. 147.<br />
The use made by Fresnel of vis viva conservation in deriving the laws of<br />
reflection, refraction <strong>and</strong> polarisation of light is explicitly mentioned, together<br />
<strong>with</strong> the application of the principle to interference, thus displaying a broad <strong>and</strong><br />
deep knowledge of physical problems <strong>and</strong> physical literature. 148<br />
The application of the principle of "Kraft" suggests that if there is a loss of vis<br />
viva due to absorption of elastic, acoustic or heat waves, a different kind of<br />
quantitatively equivalent "force" must appear 149. Heat must be produced by the<br />
absorption of caloric rays, but Helmholtz asserted that it has not yet been proved<br />
experimentally that the same quantity of heat which disappears in the radiating<br />
body reappears in the irradiated one (if losses are excluded) 150. This is a first<br />
instance of <strong>Helmholtz's</strong> approach in the last four chapters of the Erhaltung : to<br />
suggest <strong>and</strong> outline applications of the principle, independently of experimental<br />
corroborations.<br />
Energetik P.59.<br />
145 Comments on <strong>Helmholtz's</strong> definition of elasticity <strong>and</strong> on interference in: Helm<br />
146 Helmholtz Erhaltung Pp.20-1.<br />
147 Planck Princip P.39 noted <strong>Helmholtz's</strong> early acceptance of this theory. Brush<br />
showed that between 1830 <strong>and</strong> 1850 a number of physicists believed in a wave theory of heat,<br />
based on the analogies discovered by Melloni between radiant heat <strong>and</strong> light. The influences<br />
on the "energy" debates are discussed, including <strong>Helmholtz's</strong> explicit references in the<br />
Erhaltung . Brush, Stephen. "The Wave Theory of Heat: A Forgotten Stage in the Transition<br />
from the Caloric Theory to Thermodynamics". In BJHS 5 (1970-71): 145-67. Reprinted in:<br />
Brush, Stephen. The Kind of Motion We Call Heat. 2vols. Amsterdam: North Holl<strong>and</strong>, 1976.<br />
Vol.2 pp. 303-25. .<br />
148 Helmholtz Erhaltung P.23.<br />
149 Helmholtz Erhaltung P.23.<br />
150 Helmholtz Erhaltung P.24
Helmholtz, while asserting that light absorption can cause heat, light<br />
(phosphorescence) <strong>and</strong> chemical effects, identified light <strong>with</strong> radiations<br />
producing thermal <strong>and</strong> chemical effects 151.<br />
The chapter ends <strong>with</strong> some remarks on the (small) effects of light <strong>and</strong> chemical<br />
rays on the eye, an indication perhaps of a small value for their heat equivalent 152.<br />
The quantitative relations of the chemical effects produced by light were not well<br />
known, <strong>and</strong> Helmholtz believed that relevant magnitudes were only involved in<br />
the case of light absorbed by the green parts of plants 153.<br />
Even in this short chapter, which mostly recalls already known results,<br />
<strong>Helmholtz's</strong> method started to reveal its fruitfulness in organising a very large<br />
amount of physical knowledge. But some limits also became evident: the<br />
difficulty of identifying the "tension forces" reduces the conservation of "force" to<br />
a correlation principle here. Moreover, it is evident that the principle can only be<br />
heuristically fruitful if its predictions are supported by already existing empirical<br />
laws in the various fields referred to. Even in this case, for the corroboration of<br />
the principle, the knowledge of specific coefficients of equivalence is necessary,<br />
but Helmholtz lacked this knowledge. His efforts had thus to be confined to<br />
broad theoretical applications based on his surprisingly deep knowledge of the<br />
physical literature.<br />
5 Force equivalent of heat: a theoretical approach<br />
The fourth chapter of <strong>Helmholtz's</strong> Erhaltung deals <strong>with</strong> the force equivalent of<br />
heat. It would be easy to believe this to be the central part of the essay, but in fact<br />
it is not. It is, rather, the chapter in which most notably the differences <strong>with</strong> the<br />
papers of Mayer (unknown to Helmholtz) <strong>and</strong> Joule (known) become evident.<br />
Surprisingly, Helmholtz, at variance <strong>with</strong> the other two, did not establish <strong>with</strong><br />
accuracy the mechanical equivalent, <strong>and</strong> did not seem concerned by the problem.<br />
He was more interested in a theoretical interpretation of the thermal phenomena<br />
151 Helmholtz Erhaltung P.24. He quoted Melloni <strong>and</strong> Brucke, always <strong>with</strong><br />
references in Poggendorff's Annalen .<br />
152 Helmholtz Erhaltung P.24.<br />
153 Helmholtz Erhaltung P.25.
through his own framework than in a specific determination of the equivalent. In<br />
this long chapter the use of the formalism is limited to a discussion of Clapeyron's<br />
<strong>and</strong> Holtzmann's laws <strong>and</strong> the whole approach is rather qualitative. Strangely<br />
enough, the results here will not be as far-reaching as in the next two chapters.<br />
The lack of a determination of the mechanical equivalent in this fourth chapter<br />
probably justifies <strong>Helmholtz's</strong> later claim 154 that the Erhaltung was meant more<br />
as a review <strong>and</strong> a synthesis of contemporary physical knowledge than to produce<br />
original experimental results.<br />
Helmholtz started applying his method of looking for actual compensations<br />
(equivalents) to an apparent loss of "force" (energy). Through his own principle<br />
of conservation of force he identified the compensation for the loss of living<br />
force in inelastic collision <strong>and</strong> in friction <strong>with</strong> a supposed increase of tension<br />
forces, namely internal elastic forces, due to the variation of "the molecular<br />
constitution of the bodies" 155 <strong>and</strong> <strong>with</strong> acoustical, thermal <strong>and</strong> electrical effects.<br />
First comes the case of collision of non-elastic bodies: the vis viva lost must<br />
reappear as an increase of elastic forces (tension forces), as heat <strong>and</strong> sound. In<br />
the case of friction instead we have increase of elastic forces, heat <strong>and</strong> electricity.<br />
If we do not take into account molecular effects <strong>and</strong> electricity, we can pose two<br />
great problems: a) does a loss of vis viva correspond to an equivalent amount of<br />
heat?; <strong>and</strong> then: b) how can we frame heat in a mechanical interpretation? 156 The<br />
first question is connected to a "correlation" approach, the second one is specific<br />
to <strong>Helmholtz's</strong> program.<br />
Surprisingly the first question is disposed of very rapidly, in a few lines, <strong>and</strong><br />
<strong>with</strong>out definite results.<br />
Helmholtz, who did not know of Mayer's <strong>and</strong> Colding's research, asserted that to<br />
this problem "perhaps" too few efforts have been dedicated. He actually only<br />
cited a paper of Joule (1845) 157 <strong>and</strong> recalled his attempts at establishing a<br />
mechanical equivalent through the heat produced by the friction of water in<br />
narrow tubes <strong>and</strong> in vessels (the famous paddle wheel experiment). Helmholtz<br />
asserted that Joule's result in the first case is that the heat needed to raise by 1°C<br />
154 Helmholtz WA 1 P.74.<br />
155 Helmholtz Erhaltung P.26.<br />
156 Helmholtz Erhaltung P.27<br />
157 Joule, James P. "On the Existence of an Equivalent Relation between Heat <strong>and</strong> the<br />
Ordinary Forms of Mechanical Power." In Phil. Mag ser3, XXVII, p.205, reprinted in<br />
Scientific Papers Pp.202-5 , signed August 6,1845.
the temperature of 1 Kg of water can raise 452 Kg to 1 meter, <strong>and</strong> 521 Kg in the<br />
second case 158. <strong>Helmholtz's</strong> judgement on Joule's work, the only original<br />
experimental determination of the mechanical equivalent of heat cited in the<br />
Erhaltung, is very severe. For Helmholtz, Joule's measurements are not adequate<br />
to the "difficulties of the research" <strong>and</strong> thus it is not possible to accept the results<br />
as correct: "probably the figures are too high". <strong>Helmholtz's</strong> criticism of the only<br />
empirical evidence corroborating his own theoretical approach is surprising.<br />
Koenigsberger later asserted 159 that he had read Joule's papers just before<br />
publishing his own essay, but nevertheless, being a good experimentalist <strong>and</strong><br />
being involved in experimental physiological research in a closely related subject,<br />
the judgement of inaccuracy referred to experiments that were indeed very<br />
accurate requires an explanation.<br />
Tyndall, trying to provide such an explanation in 1853, in a note 160 to the English<br />
translation of the Erhaltung , warned the reader that Helmholtz was only<br />
acquainted <strong>with</strong> Joule's early works; Tyndall again in 1863 161, during the famous<br />
controversy <strong>with</strong> Tait <strong>and</strong> Thomson on the Mayer or Joule priority, quoted<br />
<strong>Helmholtz's</strong> remarks <strong>and</strong> to justify them asserted that Joule's measurements of the<br />
equivalent in 1843 162 varied between 1040 <strong>and</strong> 587 foot pounds. But actually<br />
Helmholtz, in the passage of the Erhaltung we are dealing <strong>with</strong>, did not refer to<br />
Joule's 1843 but to his 1845 paper. In this paper Joule gave the results of the<br />
paddle wheel experiments (890 foot pounds) <strong>and</strong> recalled his previous results 163:<br />
823 fp. in 1843 from magnetoelectrical experiments, 795 fp. in 1845 from the<br />
158 For a discussion of these figures see below.<br />
159 Helmholtz "On the Interaction of Natural Forces <strong>and</strong> recent Physical Discoveries<br />
bearing on the same" Phil Mag XI, 1856:489-518, on p.499; see also Helmholtz<br />
Autobiographical Sketch P.12; Koenigsberger HvH P.44.<br />
160 Helmholtz, Hermann. "On the <strong>Conservation</strong> of Force; a Physical Memoir."<br />
Trans.John Tyndall. In Scientific Memoirs-Natural Philosophy , Tyndall <strong>and</strong> Francis (eds.),<br />
vol. I, p.II, London,1853 :114-162; P.131.<br />
161 Tyndall, John. Philosophical Magazine 1863 Pp. 375-6<br />
162 Joule, James. "On the Caloric Effects... " 1843. Repr in The Scientific papers<br />
Vol.1. Pp.123-159. Pp.151 <strong>and</strong> 153.<br />
163 Joule "Equiv Relat" quoted n.244, P.204 repr.
arefaction of the air 164, 774 fp. from unpublished experiments on the friction of<br />
water moving in narrow tubes 165. Joule averaged the two experiments resulting<br />
from the friction of water (890 <strong>and</strong> 774) to 832 <strong>and</strong> again averaged the results of<br />
the three distinct classes of experiments mentioned (823,795 <strong>and</strong> 832) to 817.<br />
The results are then far more precise than indicated by Tyndall <strong>and</strong> such that<br />
<strong>Helmholtz's</strong> criticisms seem excessive (the final accepted value of the equivalent<br />
was to be 778).<br />
The fact that Helmholtz strongly criticised in 1847 the only experimental<br />
evidence he knew in favour of his own approach deserves a better explanation<br />
than Tyndall's. In my view three elements influenced Helmholtz: a) the Erhaltung<br />
is a theoretical work, whose origin is largely independent from experimental<br />
results: it was really meant to be a reinterpretation of already existing knowledge,<br />
<strong>and</strong> thus its value did not have to be connected <strong>with</strong> "doubtful" experimental<br />
results, i.e. results that did not enjoy universal recognition. Joule, an amateur<br />
scientist, in 1847 still had serious problems in getting scientific recognition <strong>and</strong><br />
Helmholtz might not have wanted to rely on such an ally; b) Helmholtz quotes<br />
Joule four times in different passages of the fourth chapter, but despite this it is<br />
possible that he became aware of Joule's papers only in the final preparation of<br />
the Erhaltung : Joule in fact is not quoted in the "Bericht" (written in October<br />
1846). Thus Helmholtz might not have managed to master his results; c) the third<br />
component of my explanation is of a completely different sort. In reworking out<br />
the conversions between the British <strong>and</strong> the continental units (from Fahrenheit<br />
degrees, feet <strong>and</strong> pounds to centigrade degrees, kilograms <strong>and</strong> meters) I found<br />
that Helmholtz made a systematic mistake in the conversions.<br />
164 Joule, James. "On the Changes of Temperature produced by the Rarefaction <strong>and</strong><br />
Condensation of Air." In Phil.Mag ser.3 May 1845. Repr. in The Scientific Papers. Vol.1.<br />
Pp.172-189.<br />
165 The two first figures (823, 795) given by Joule in "Equiv Relat" of 1845 at p. 204<br />
of the reprint do not agree <strong>with</strong> his own values in the papers referred to: "Caloric Effects" of<br />
1843 <strong>and</strong> "Condens <strong>and</strong> Raref" of 1845. The value of the magnetoelectric experiments is 838<br />
(see repr. P.156 <strong>and</strong> 187); the value of the air experiments is 798 (see P.187); the two values<br />
of 823 <strong>and</strong> 795 correspond instead to the two series of experiments mentioned at Pp.179-180<br />
<strong>and</strong> P.187 of the reprint of the paper on "Condens <strong>and</strong> Raref" <strong>and</strong> discarded from the values<br />
giving the final average of 798. The 1845 value of 774 for narrow tubes was 770 in 1843: see<br />
repr. P.157.
In his search for a mechanical equivalent of heat Joule equated the quantity of<br />
heat needed to increase 1 lb of water by 1 °F (later to be called BTU) <strong>with</strong> the<br />
corresponding work expressed in feet by force pounds. From his experiments<br />
<strong>with</strong> the friction of water he obtained a mechanical equivalent of 890 ft lb/BTU<br />
for vessels <strong>and</strong> 774 ft lb/BTU for narrow tubes.<br />
Considering the conversion factors between British <strong>and</strong> continental<br />
units (BTU=0.252 kcal; ft lb=O.1382 kgm) to express Joule results in kgm/kcal<br />
we have to multiply them by a factor of 0.5486 (= 0.1382/0.252). Joule's values<br />
of 890 <strong>and</strong> 774 thus correspond to 488 <strong>and</strong> 424. The second figure is very close<br />
to 778, the later accepted final value of the mechanical equivalent of heat (in<br />
continental units: 778 x 0.5486= 427 kgm/kcal).<br />
The results of <strong>Helmholtz's</strong> conversion, as recalled, are different:<br />
instead of the values of 488 <strong>and</strong> 424 he gives 521 <strong>and</strong> 452 for vessels <strong>and</strong> narrow<br />
tubes respectively.<br />
Helmholtz was thus right in asserting that the conversion values of Joule's<br />
experiments given in the Erhaltung were too high, but wrong in asserting that<br />
this was Joule's fault <strong>and</strong> that Joule's experiments were not accurate. The mistake<br />
was introduced by Helmholtz himself <strong>and</strong> it is relevant both for his general<br />
evaluation of Joule <strong>and</strong> for the specific comparison (made below in the same<br />
chapter) of Joule's results <strong>with</strong> Holtzmann's results (already given in metric<br />
units). In 1881 in a footnote, Helmholtz modified his comments <strong>and</strong> praised<br />
Joule's work but did not acknowledge his own conversion mistake of 1847 166.<br />
We may wonder why such a careful researcher made a mistake in the conversion.<br />
The mistake is a systematic one: 521/488.3 = 1.067; 452/424.6 = 1.065. My<br />
answer to this question is that Helmholtz was using the French foot 167, a wellknown<br />
unit of measurement, which is equal to 12.8 inches, that is, 0.3251 m. In<br />
fact 12.8/12 = 1.067.<br />
Joule's researches thus did not get the appreciation they deserved in the<br />
Erhaltung ; surprisingly, Helmholtz left the problem unsettled ( he dedicated a<br />
few lines to the whole discussion of the mechanical equivalent) <strong>and</strong> rapidly<br />
turned to the second question outlined above, the theoretical one, much more<br />
166 Helmholtz WA 1 P.33.<br />
167 On the French foot: Jerrard,H.G. <strong>and</strong> McNeill, D.B. A Dictionary of Scientific<br />
Units . 5th ed. Chapman <strong>and</strong> Hall, 1986, P.45.
important in his view: "to what extent can heat correspond to a force<br />
equivalent"? 168<br />
Here force equivalents should not be confused <strong>with</strong> mechanical equivalents: the<br />
firsts are theoretically identifiable "energy terms"; the second are numerical<br />
conversion factors.<br />
The caloric theory was discussed in explicit reference to the interpretation of<br />
Carnot <strong>and</strong> Clapeyron 169, for whom the force equivalent is the work produced in<br />
the passage of a certain amount of caloric from a higher to a lower temperature.<br />
Helmholtz criticised 170 W.Henry's <strong>and</strong> Berthollet's interpretation of the heat<br />
produced by friction as displacement of caloric <strong>and</strong> asserted that results from the<br />
field of electricity showed that the total amount of the heat of a body can be<br />
actually increased. Helmholtz recalled experimental evidence, entirely based on<br />
research in electricity, against the caloric theory of heat <strong>and</strong> in favour of the<br />
mechanical one. While frictional <strong>and</strong> voltaic electricity did not give<br />
unquestionable evidence, because the heat produced could be interpreted as<br />
caloric displaced, "we still have to explain in a purely mechanical way the<br />
production of electrical tensions in two processes, in which any quantity of heat<br />
that can be assumed as transferred never appears" 171 : electrical (not<br />
electromagnetic) induction <strong>and</strong> movements of magnets. Helmholtz gave the<br />
example of an electrophorus used to charge a Leyden jar for the first, <strong>and</strong> for the<br />
second the example of electromagnetic machines where "heat can be developed<br />
ad infinitum" 172. It is only at this stage that Helmholtz recalled Joule's experiments<br />
of 1843 173 <strong>and</strong> asserted that Joule "endeavoured to show directly" that the<br />
electromagnetic current produced heat <strong>and</strong> not cold even in that part which is<br />
under the actual action of the magnet (no displacement of caloric is thus<br />
conceivable in the electrical circuit). Again the minor role that Joule's results<br />
play in the exposition of <strong>Helmholtz's</strong> ideas must be noted .<br />
For Helmholtz thus the caloric theory must be rejected, because heat can be<br />
produced indefinitely through mechanical <strong>and</strong> electrical forces, <strong>and</strong> the<br />
mechanical theory must take its place. As seen in the previous section, this result<br />
168 Helmholtz Erhaltung P. 27.<br />
169 Helmholtz Erhaltung P.28.<br />
170 Helmholtz Erhaltung P.28<br />
171 Helmholtz Erhaltung P. 29.<br />
172 Helmholtz Erhaltung P.30.<br />
173 Helmholtz Erhaltung P.30 ( Joule's rezsults were wrongly dated 1844)
had already been achieved in the Bericht. What is new <strong>and</strong> specific of the<br />
Erhaltung is the application of the theoretical framework of tension <strong>and</strong> living<br />
forces to the mechanical theory of heat. This again is done in a purely conceptual<br />
<strong>and</strong> qualitative way, <strong>with</strong>out mathematical formulation: free heat will now be<br />
interpreted as the quantity of living force of thermal movement <strong>and</strong> latent heat as<br />
the quantity of tension forces, namely elastic forces of atoms. Helmholtz quoted<br />
Ampère while making an attempt at clarifying the atomistic view of nature <strong>and</strong><br />
the way in which atomic movements can explain radiation <strong>and</strong> conduction. But<br />
the whole subject is highly speculative <strong>and</strong> Helmholtz is satisfied <strong>with</strong> "the<br />
possibility that thermal phenomena be conceived as motions" 174. In fact in this<br />
case the conservation of force "could be verified" whenever conservation of<br />
caloric substance was supposed to hold.<br />
In my view, <strong>Helmholtz's</strong> cautiousness is completely justified. It is not due to a<br />
lack of conceptual clarity, but to the awareness of the lack of empirical<br />
corroboration. In my opinion, the doubt raised by Helm whether, at this point of<br />
the Erhaltung , Helmholtz was still "uncertain <strong>and</strong> vacillating" on the validity of<br />
the impossibility of the perpetuum mobile outside mechanics 175 is to be rejected.<br />
Despite being qualitative, <strong>Helmholtz's</strong> conceptual scheme is wide- ranging. To<br />
explicitly conceive atoms as possessing not only living forces but also tension<br />
forces is a bold step forward <strong>and</strong> the analogy <strong>with</strong> free <strong>and</strong> latent heat seems very<br />
well defined. It allowed in fact an easy reinterpretation of previous knowledge in<br />
the new terms.<br />
The reinterpretation of the heat produced in chemical processes follows: Hesse's<br />
law "partially verified also by experience" had been deduced from the caloric<br />
theory. It asserts that "the heat developed in the production of a chemical<br />
compound is independent of the order <strong>and</strong> the intermediate steps of the<br />
process" 176. The law, as shown in the "Bericht", is in agreement <strong>with</strong> the forceequivalent<br />
hypothesis (correlation principle), but here Helmholtz showed that it<br />
can be interpreted in terms of the new concepts of living <strong>and</strong> tension forces: the<br />
heat produced is now considered a living force, generated by the chemical forces<br />
174 Helmholtz Erhaltung P.31.<br />
175 Helm hinted that <strong>Helmholtz's</strong> cautiousness in the application of his second root<br />
(central forces <strong>and</strong> mechanical hypothesis) to thermal phenomena implies that he had doubts<br />
on the first one (impossibility of perpetual motion) as well, <strong>and</strong> thus on the whole conservation<br />
problem. G.Helm Energetik P.44<br />
176 Helmholtz Erhaltung P.32.
of attraction that play the role of tension forces. Here, implicitly, Helmholtz<br />
applied the mechanical concept of conservation developed in the first chapter: the<br />
vis viva developed between two definite configurations of the system is<br />
independent of the trajectory.<br />
Helmholtz then turned to the final problem of the chapter: the disappearance of<br />
heat. Very little attention had been dedicated to this problem (in correspondence<br />
<strong>with</strong> the little attention dedicated to the ad nihilum nil fit) :<br />
"As yet nobody has inquired if heat disappears in the production of mechanical<br />
force: which would be a corollary of the law of conservation of force" 177.<br />
In discussing this problem again, Helmholtz revealed his inclination towards a<br />
theoretical approach: both the transformations of work into heat <strong>and</strong> heat into<br />
work were assumed, but as necessary consequences of the principle <strong>and</strong> not on<br />
the basis of experimental results.<br />
Helmholtz quoted Joule 178 for the third time, asserting that his results on this topic<br />
were the only ones available <strong>and</strong> that they seem "sufficiently reliable" 179. The<br />
experience referred to is a famous one: compressed air exp<strong>and</strong>ing against air<br />
pressure cools down; this does not take place if the air exp<strong>and</strong>s in a vacuum. In<br />
the first case the compressed air has to exert a mechanical force to overcome the<br />
resistance of the air pressure <strong>and</strong> in the second case it does not. Thus in the first<br />
case the heat which has disappeared can be equated to the work done <strong>and</strong> thus a<br />
mechanical equivalent can be found, but no equivalent is mentioned here 180.<br />
Finally comes one of the most puzzling sections of the chapter: the discussion of<br />
the research of Clapeyron <strong>and</strong> Holtzmann. Helmholtz was aware that both<br />
researchers had been working on the assumptions of the caloric theory, in fact in<br />
the "Bericht" he asserted that they dealt <strong>with</strong> the propagation more than the<br />
production of heat. But here Helmholtz spoke of their research as "tending to find<br />
out the force equivalent of heat" 181 <strong>and</strong> compared them <strong>with</strong> his own. Clapeyron's<br />
177 Helmholtz Erhaltung P.33.<br />
178 Joule "Condens <strong>and</strong> Raref" Phil Mag P.379, rep. P.184.<br />
179 Helmholtz Erhaltung P.33.<br />
180 At Joule's <strong>and</strong> <strong>Helmholtz's</strong> ignorance this same approach had been followed,<br />
<strong>with</strong>out actually performing new experiences but reinterpreting in original way old data, by<br />
Mayer in 1842. Mayer had worked out an equivalent of 365 Kgm. If the data Mayer utilised<br />
had been more reliable ( for instance the ones of Regnault) his value would have been correct:<br />
425 Kgm, see Haas Entwickl P.87.<br />
181 Helmholtz Erhaltung P. 33.
approach is discussed at length <strong>and</strong> criticised: only for gases the law established<br />
by Clapeyron on the assumptions of the caloric theory had received empirical<br />
support, but in the case of gases it is equivalent to Holtzmann's.<br />
Holtzmann assumed that if a certain quantity of heat "enters" in a gas it either<br />
produces an increase of temperature or an expansion. In this second case the<br />
work done gives, following <strong>Helmholtz's</strong> account of Holtzmann, the possibility to<br />
calculate the mechanical equivalent of heat. On the grounds of Dulong's values<br />
for the specific heats of gases, Holtzmann's equivalent is 374 Kgm 182. Helmholtz<br />
warned that this could be accepted in the framework of the conservation of force<br />
only if all the living force of the heat communicated was actually given as work,<br />
that is, if the sum of the living forces <strong>and</strong> the tension forces, or, in the old<br />
terminology, the quantity of free <strong>and</strong> latent heat, of the exp<strong>and</strong>ed gas was the<br />
same as the one of the denser gas at the same temperature. This approach is in<br />
agreement <strong>with</strong> the above quoted one of Joule, <strong>and</strong> Helmholtz compared the<br />
equivalents of Holtzmann <strong>and</strong> Joule. The 374 Kgm of the former are compared<br />
<strong>with</strong> a series of results of the latter, who is credited <strong>with</strong> having actually<br />
performed the experiments <strong>and</strong> not to have only reinterpreted previous data. Five<br />
values were given from Joule's results 183: two are the already recalled ones of 452<br />
<strong>and</strong> 521 (which, as already explained, should be 424 <strong>and</strong> 488) deriving from the<br />
friction of water, <strong>and</strong> three (481, 464, 479 which should be 451, 435, 449) are<br />
quoted <strong>with</strong>out reference. My view is that these three values are taken from<br />
Joule's 1845 paper: "On the Existence of an Equivalent Relation". The first<br />
(481/451) refers to the 1843 experiments <strong>with</strong> an electromagnetic engine, the<br />
second (464/435) to the 1845 experiments on air referred to above, <strong>and</strong> the third<br />
one to the final average mentioned in Joule's paper 184. As noted above, the<br />
comparison of the results of the two researchers is deeply distorted by the<br />
systematic error in the conversion of Joule's units of measurement. The chapter<br />
ends <strong>with</strong> a detailed discussion <strong>and</strong> comparison, which includes a table, of the<br />
laws of Clapeyron <strong>and</strong> Holtzmann 185.<br />
182 Helmholtz Erhaltung P. 35.<br />
183 Helmholtz Erhaltung P. 36.<br />
184 Going back from <strong>Helmholtz's</strong> values to British ( <strong>and</strong> not French) foot-pounds we<br />
get 822, 793, 819 that are reasonably close to Joule's figures of 823, 795, 817 respectively.<br />
185 Helmholtz Erhaltung P.37. Helmholtz was later to claim priority in the<br />
interpretation of Carnot's function: see Wolff "Clausius".
On this point Clausius, who had given 186 in 1850 a mechanical equivalent of 421<br />
Kgm, in 1854 moved a serious objection, asserting that Helmholtz had<br />
misunderstood Holtzmann's law, in which the caloric concept played a role which<br />
cannot be eliminated. This was among the very few criticisms that Helmholtz<br />
accepted during the controversy 187.<br />
In 1882 in an appendix to this chapter, Helmholtz discussed the priority<br />
problem 188.<br />
6 Electricity, galvanism <strong>and</strong> thermo-electric currents: Helmholtz <strong>and</strong> the<br />
batteries<br />
The fifth <strong>and</strong> the longest chapter of the Erhaltung is dedicated to applying the<br />
law of conservation of force to static electricity, galvanism <strong>and</strong> thermo-electric<br />
currents 189.<br />
Once again Helmholtz displayed an extraordinarily detailed knowledge of<br />
empirical laws of physics in his attempt at further theoretical applications of his<br />
principle.<br />
The first application is an easy one: Coulomb's law, being a strictly central force<br />
law <strong>with</strong> attractive <strong>and</strong> repulsive forces, offers the best possible example of how<br />
to formulate a sum of tension forces, to be equated <strong>with</strong> an increase of vis viva.<br />
But the difficulties do not wait for long: Helmholtz, explicitly referring to Gauss<br />
186 Clausius, Rudolf "Ueber die bewegende Kraft der Wärme und die Gesetze, welche<br />
sich daraus für die Wärmelehre selbst ableiten lassen" Annalen 79 (1850): 368-97, 500-24.<br />
187 Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In<br />
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, pp.76-93. P.90. Still Truesdell believes that<br />
<strong>Helmholtz's</strong> approach in 1847, while wrong in other aspects, in this was consistent: Truesdell,<br />
Clifford. The Tragicomic History of Thermodynamics. 1822-1854. New York: Springer,1980,<br />
P.162; Truesdell believes that the Erhaltung is "<strong>Helmholtz's</strong> weakest work" ibid. P.161.<br />
188 Helmholtz WA 1 Pp.71-4.<br />
189 Given that most of the chapter (from P. 48 to P. 58) deals <strong>with</strong> a careful analysis<br />
of batteries, Kuhn's remark (Kuhn Sim Disc P.73) that Helmholtz did not discuss batteries in<br />
his Erhaltung is difficult to underst<strong>and</strong>.
<strong>and</strong> unaware of Green's results190, introduced the concept of electric potential. He<br />
defined the quantity<br />
⎛ eieii − ⎝ r ⎠ ⎞<br />
corresponding to the sum of the tension forces consumed, <strong>and</strong> also of course to<br />
the living forces acquired in the motion of the two charges from an infinite<br />
distance to the distance r, as the potential of the two electrical elements for the<br />
distance r 191.<br />
The principle of conservation can thus be expressed in a new way: "the increase<br />
of vis viva in whichever movement must be considered equal to the difference of<br />
the potential at the end of the trajectory respect to the potential at the<br />
beginning" 192 (the sign of the potential is opposite to that of modern convention).<br />
The potential as defined by Helmholtz is equivalent but for the sign to what later<br />
was to be called potential energy. He showed a good grasp in setting the<br />
relations between potential <strong>and</strong> work, for instance in the case of the potential of<br />
one body <strong>with</strong> respect to another193. Here in fact he precisely stated the<br />
equivalence between potential <strong>and</strong> work. But problems occur in the definition of<br />
the potential of a body on itself (the sum of the potentials of an electric element<br />
of a body <strong>with</strong> respect to all the other electrical elements of the same body): in<br />
this case <strong>Helmholtz's</strong> definition is double the modern convention, but what mostly<br />
matters, does not correspond to the work done194 (the potential is supposed to be<br />
double the work done). This shows the "independence" of the two concepts in<br />
<strong>Helmholtz's</strong> approach. In the original 1847 edition of the Erhaltung there is a<br />
final correction (the only one) referring exactly to these problems. This is a clear<br />
indication of the incertitudes <strong>and</strong> difficulties faced by Helmholtz <strong>with</strong> concepts<br />
that in 1847 were by no means common195. Helmholtz in 1882 196 acknowledged<br />
190 According to Koenigsberger, Helmholtz knew of Green's theorem later, in<br />
Koenigsberg, partially through F.Neumann: Koenigsberger H v H P.100.<br />
Erhaltung<br />
191 Helmholtz Erhaltung P.38.<br />
192 Helmholtz Erhaltung P.39.<br />
193 Helmholtz Erhaltung Pp.39-40<br />
194 Helmholtz Erhaltung Pp.42-3. Clausius was the first to criticise this part of the<br />
195 In the 1882 reprint of the Erhaltung in WA 1 the correction is incorporated in the<br />
text (as in the English translations of Tyndall, Kahl <strong>and</strong> Lindsay), <strong>and</strong> Helmholtz warned the
in a note that he had considered twice every interaction of two electrical particles<br />
( most contemporary textbooks warn the students not to make such a 'mistake' 197)<br />
<strong>and</strong> that the use of other authors to equate potential <strong>and</strong> work was more<br />
appropriate. Still he maintained that his 1847 approach in relating potential <strong>and</strong><br />
work was basically correct.<br />
In my view <strong>Helmholtz's</strong> claim is based on reasonable grounds: I believe he was<br />
among the first to interpret <strong>and</strong> use "correctly" the "new" mathematical tool (the<br />
potential) in physical research 198.<br />
Helmholtz thus unified explicitly in this essay the tradition of analytical<br />
mechanics (potential function of Gauss, Hamilton <strong>and</strong> Jacobi) <strong>and</strong> of engineering<br />
(concept of work), but it is noticeable that, at variance <strong>with</strong> Clausius' more<br />
mathematical approach of 1852 199, he arrived at the concept of potential not<br />
through the concept of work as a total differential, but straight from the concept<br />
of sum of tension forces. The fact that theoretical rather than mathematical<br />
physics was at the root of <strong>Helmholtz's</strong> approach is evident from his attempt to<br />
clarify the "mathematical" potential through the introduction of physically sound<br />
concepts <strong>and</strong> not viceversa. Helmholtz introduced 200 first the "equilibrium<br />
surfaces", later identified <strong>with</strong> equipotential surfaces, <strong>and</strong> second the "free<br />
tension", later to be identified by Helmholtz himself <strong>with</strong> the mathematical<br />
reader <strong>with</strong> a note to the last appendix WA 1 P.75 (the latter in Kahl's translation is simply<br />
omitted).<br />
196 Helmholtz WA 1 P.75<br />
197 For instance: Feynman, Richard. The Feynman's Lectures on Physics. 3 vols.<br />
Addison Wesley, 1963; vol 2 P.8/15<br />
198 Hoppe's claim that the identification of work <strong>and</strong> potential started <strong>with</strong> Riemann is<br />
probably due, being Hoppe a committed Weberian, to the lasting effects of the Weber/<br />
Helmholtz controversy; see Hoppe,E. Histoire de la Physique. Paris: Payot, 1928; P.570.<br />
199 Clausius, Rudolf. "Ueber das mechanische Aequivalent einer elektrischen<br />
Entladung und die dabei stattfindende Erwärmung des Leitungsdrahtes" Pogg Ann 86, 1852:<br />
337-375; transl in:"On the Mechanical Equivalent of an Electric Discharge, <strong>and</strong> the Heating of<br />
the Conducting Wire which accompanies it." In Tyndall <strong>and</strong> Francis Scientific Memoirs on<br />
Natural Philosophy 1 (1853): 1-32.<br />
200 Helmholtz Erhaltung Pp.40-1 <strong>and</strong> P.42
potential function 201. It is worth noting that still in 1847 <strong>Helmholtz's</strong> idea of<br />
electric tension was reminiscent of Volta's influential "density of electricity" 202.<br />
Helmholtz <strong>with</strong> great intellectual ingenuity struggled to apply his conceptual<br />
framework of living <strong>and</strong> tension forces to every realm of nature: he did not use<br />
mathematical requirements as heuristic tools <strong>and</strong> thus differed from mathematical<br />
physicists such as Clausius <strong>and</strong> Riemann to a great extent. But he also differed<br />
widely from experimentalists such as Joule: in discussing the heat generated in an<br />
electrical discharge he stressed the interpretation of the heat as the vis viva<br />
produced by a decrease of the quantity of tension forces (i.e. of the potential) <strong>and</strong><br />
of the discharge as an oscillatory process of alternating currents, more than the<br />
search for experimental results. In fact for the mechanical equivalent of heat<br />
(whose numerical value enters in the laws expressed, given the electrostatic<br />
system of units utilised 203) he asserted that "up to now observations are lacking".<br />
Detailed information is instead given on the relation between the heat produced<br />
through the discharge of a specific battery <strong>and</strong> the shape <strong>and</strong> dimension of the<br />
connecting wire 204 (quoting Riess, Vorselman de Heer <strong>and</strong> Knochenauer).<br />
The fact that conceptual explanation played a greater role than empirical<br />
corroboration or mathematical formalism is again exemplified in the discussion of<br />
galvanism. For Helmholtz 205 the contact law of Volta, if correctly interpreted, is<br />
not in disagreement, as often remarked 206, <strong>with</strong> the impossibility of perpetual<br />
motion.<br />
201 Helmholtz, Hermann. "Ueber einige Gesetze der Vertheilung elektrischer Ströme<br />
in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche." In Pogg Ann<br />
89 (1853): 211-33, 352-77. Repr. in WA 1 p.475, P.224.<br />
202 On Volta's influence in Germany see: Teichmann, Jurgen. "L'influenza di A.Volta<br />
in Germania." Quaderni del Giornale di Fisica 3 (1977): 43-60. In 1849 Kirchhoff unified the<br />
concepts of electric tension, electrostatic <strong>and</strong> electrodynamic potential : Kirchhoff,Gustav. "On<br />
a Deduction of Ohm's Law, in Connection <strong>with</strong> the Theory of Electrostatics" Phil Mag 3<br />
XXXVII (1850): 463-8; translated from Pogg Ann LXXVIII (1849) P.506; see also : my<br />
<strong>Energy</strong> <strong>and</strong> Electr Pp.77 <strong>and</strong> 96-7; Archibald, Thomas. "Tension <strong>and</strong> Potential from Ohm to<br />
Kirchhoff." In Centaurus 31 (1988): 141-63.<br />
criticisms.<br />
203 See Planck Princip P.41.<br />
204 Helmholtz Erhaltung Pp.43-4. This was going to be another object of Clausius'<br />
205 Helmholtz Erhaltung P.45.<br />
206 Kuhn seems still to believe it: Kuhn Sim Disc P.73. See above n.20.
Volta's contact tensions in fact are not equivalent to a certain quantity of "force":<br />
they do not produce an electrical imbalance but instead originate from an<br />
electrical imbalance. For a correct interpretation of the processes it is necessary<br />
to restrict the contact law to first class conductors (metals) <strong>and</strong> realise that the<br />
second class conductors conduct only by means of an electrolytic process. Thus a<br />
contact force can be interpreted in terms of attractive <strong>and</strong> repulsive forces of the<br />
two metals which remove electrical charges of the contact area from one metal<br />
to the other. Equilibrium is reached when an electrical particle in the passage<br />
from one metal to the other does not acquire or lose living force, that is, when<br />
the variation of living force from one metal to the other is compensated by an<br />
identical variation of tension forces independently of the shape <strong>and</strong> dimension of<br />
the contact surfaces <strong>and</strong> in agreement <strong>with</strong> the galvanic series of tensions.<br />
Once more, conservation of force based on central forces gave a conceptual<br />
explicative framework. But the case was different <strong>with</strong> the long analysis of<br />
galvanic currents.<br />
<strong>Conservation</strong> of force is applied to batteries not producing polarisation,<br />
producing polarisation but not chemical decomposition <strong>and</strong> producing both. Here,<br />
as enthusiastically remarked by Helm 207, conservation is applied in the nonmechanical<br />
sense, as equivalence of numerical effects <strong>with</strong>out a reinterpretation<br />
in terms of living <strong>and</strong> tension forces. The first case, batteries <strong>with</strong>out polarisation,<br />
is the only one for which we have precise laws, experimentally corroborated.<br />
Through the laws of Ohm, Lenz <strong>and</strong> Joule, Helmholtz gave the amount of heat<br />
that must be generated in the circuit to have conservation of "energy". This heat<br />
has to be equivalent to the chemical heat that would be developed <strong>with</strong>out<br />
electrical effects <strong>and</strong> the result is that the electromotive forces of the two metals<br />
be proportional to the difference of the heat developed through their oxidation<br />
<strong>and</strong> through their combination <strong>with</strong> acids.<br />
The case of polarisation <strong>and</strong> of polarisation <strong>with</strong> chemical decomposition is<br />
discussed in detail but only in a qualitative way, i.e. not only <strong>with</strong>out the<br />
application of the conceptual scheme, but even <strong>with</strong>out quantitative indications,<br />
for the lack of reliable empirical data 208.<br />
Joule is quoted again, for his experiments intended to show the equivalence of<br />
chemical <strong>and</strong> electrical heat 209. But again his results <strong>and</strong> methods are criticised<br />
207 Helm Energetik P.44.<br />
208 Helmholtz Erhaltung Pp.51-6<br />
209 Joule Phil Mag XIX 1841 P.275; XX 1843 P.204; Helmholtz Erhaltung P.56.
<strong>and</strong> judged unreliable, despite providing evidence for a part, at least, of<br />
<strong>Helmholtz's</strong> innovative programme.<br />
Once more Helmholtz turned to his main line of thought: a conceptual<br />
explanation of electrical movements between metals <strong>and</strong> fluids through attractive<br />
<strong>and</strong> repulsive forces, in analogy <strong>with</strong> what he had already achieved in the case of<br />
contact forces. In the case of polarization currents the two metals would attract<br />
positive or negative electrical charges, respectively, till saturation. In the case of<br />
chemical decomposition there is not a stable equilibrium but a continuous<br />
process. The velocity of the process does not continually increase for the loss of<br />
vis viva through development of heat. An equivalence can be derived between the<br />
heat produced (living force) <strong>and</strong> the consumption of chemical elastic force<br />
(tension force). Thus <strong>Helmholtz's</strong> conservation of "energy" helped to clarify<br />
another difficult topic.<br />
Finally, Helmholtz discussed thermo-electric currents <strong>and</strong> the Peltier effect.<br />
Without applying the concepts of tension <strong>and</strong> living forces he utilized the<br />
principle of conservation to derive two consequences (on the heat produced <strong>and</strong><br />
absorbed at equal (constant) temperatures <strong>and</strong> equal currents) but "I do not<br />
know, so far, experimental measurements for these two consequences" 210.<br />
7 Which force equivalents for magnetism <strong>and</strong> electromagnetism?<br />
In the last chapter <strong>Helmholtz's</strong> approach reveals 211 all its fertility <strong>and</strong> limits <strong>and</strong><br />
here the intrinsic difficulties connected <strong>with</strong> the formulation <strong>and</strong> application of<br />
the principle of conservation of "energy" can be better understood.<br />
In the section dedicated to magnetism the pattern follows the one for<br />
electrostatics: the inverse square law provides an easy expression for tension<br />
forces. Living forces <strong>and</strong> potentials are defined, both for two bodies <strong>and</strong> for a<br />
body on itself. An interesting application is the one for a non-magnetised steel bar<br />
brought close to a magnet, magnetised <strong>and</strong> separated. There is an expenditure of<br />
-1/2W in mechanical work (again the potential on itself W is twice the work )<br />
that is acquired by the now magnetised bar 212.<br />
210 Helmholtz Erhaltung P.60.<br />
211 Helmholtz Erhaltung P.60-9.<br />
212Helmholtz Erhaltung P.62-3.
The section on electromagnetism starts <strong>with</strong> a concise <strong>and</strong> straightforward<br />
assessment of the state of art. It shows <strong>Helmholtz's</strong> complete mastery of the<br />
subject <strong>and</strong> outlines the programme of research that he was to pursue for forty<br />
years. His statements are fundamental, then, not only for the history of energy<br />
conservation but also for the history of electromagnetism.<br />
Helmholtz reveals himself to be familiar not only <strong>with</strong> the celebrated laws of<br />
Ampère but <strong>with</strong> the more recent ones of Weber, Lenz, F.Neumann <strong>and</strong><br />
Grassman. The characterization of the different approaches is very precise:<br />
Weber's law, at variance <strong>with</strong> Ampère's, explains electromagnetic induction, but<br />
in a conceptual framework at odds <strong>with</strong> <strong>Helmholtz's</strong>. Weber in fact assumed<br />
forces depending on velocities <strong>and</strong> accelerations. Helmholtz remarked that "up to<br />
now " 213 it had not been possible to refer this law to the central force hypothesis.<br />
Both the laws of Neumann <strong>and</strong> Grassman agree <strong>with</strong> Weber's law for closed<br />
currents, the only currents for which experiments were available 214.<br />
<strong>Helmholtz's</strong> plan was thus to confine the application of the principle to closed<br />
currents <strong>and</strong> to show that the "same laws" could be deduced through the<br />
principle 215. The strategy is clear: lacking a central force law for<br />
electromagnetism, if some "empirical" laws already deduced from non -<br />
Newtonian force hypotheses could be rededuced from the principle of<br />
conservation of "energy", strong evidence would be acquired for the justificatory<br />
power of the principle. Moreover, if new consequences could be successfully<br />
predicted, its heuristic power would gain evidence.<br />
But some not minor difficulties are implied in the process; two question are<br />
relevant: a) is the dichotomy of tension <strong>and</strong> living forces really needed for<br />
<strong>Helmholtz's</strong> specific application of the principle to electromagnetic phenomena or<br />
a correlation principle (quantitative equivalence) would be sufficient?; b) does<br />
the principle offer a reliable key to discover all the force equivalents that have to<br />
be taken into account?<br />
213 Helmholtz Erhaltung P.63.<br />
214 This same remark was the starting point of the famous paper of 1870, when<br />
Maxwell's approach too was analysed: Helmholtz, Hermann. "Ueber die<br />
Bewegungsgleichungen der Elektricität für ruhende leitende Körper." In Borchardt's Journ 72<br />
(1870): 57-129; repr. in WA 1 pp.545-628. <strong>Helmholtz's</strong> suggestion to compare the different<br />
laws in the case of open currents finally led Hertz to devise <strong>and</strong> perform his famous<br />
experiments in the late eighties.<br />
215 Helmholtz Erhaltung P.64.
The relevance of the first question can be better understood if we remember that <strong>Helmholtz's</strong><br />
sharp distinction between a positional <strong>and</strong> a kinetic energy was the result of the privilege he<br />
had attributed to central Newtonian forces. It was not a result of the other basic hypothesis of<br />
the impossibility of peprpetual motion. Thus alternative expression of the principle of<br />
conservation were possible <strong>and</strong> were in fact given (correlation, generalised potentials). In<br />
which way then was <strong>Helmholtz's</strong> approach superior? Were electromagnetic phenomena better<br />
explained if interpreted in terms of the two forms of energy?<br />
The second question does not deal <strong>with</strong> the superiority of one or the other<br />
expression of the principle but <strong>with</strong> the possibility of the theoretical principles to<br />
provide new knowledge; in a way it questions the basis of the new theoretical<br />
physics: can the energy terms in a specific situation be provided by a theoretical<br />
analysis based on the principle alone or is experimental knowledge of that<br />
situation needed for a correct application of the principle? In this second case<br />
what is the advantage of using the principle?<br />
An answer to these questions can be obtained through a discussion of two of the<br />
four cases analysed in this section of the Erhaltung.<br />
The first case discussed refers to a system consisting in a magnet moving under<br />
the effect of a current 216. Helmholtz identified the tension forces <strong>with</strong> the ones<br />
utilised in the current : aAJdt (in mechanical units, <strong>with</strong> a = mechanical<br />
equivalent of heat, A = the electromotive force of a single cell, J= the current),<br />
that is, (following the result of the section on galvanism in the previous chapter),<br />
<strong>with</strong> the heat generated in the chemical process inside the battery. The living<br />
force is supposed to be composed of two parts: one, as before, is the heat<br />
generated in the circuit (W= the resistence of the circuit) by the current:<br />
the other is the living force acquired by the magnet under the effect of the<br />
current. This is identified <strong>with</strong>:<br />
J dV<br />
dt dt<br />
where V is the potential of the magnet towards the conductor carrying a unitary<br />
current (accepting Ampère's view that the electrodynamic effects of a closed<br />
current are equivalent to those of a magnetic double layer). Thus:<br />
A−<br />
J =<br />
1<br />
a dV<br />
dt<br />
W<br />
The term<br />
1<br />
a<br />
dV<br />
dt<br />
216 Helmholtz Erhaltung Pp.64-5.
was interpreted by Helmholtz as a new electromotive force: the force of the<br />
induced current. The law is similar to Neumann's, but more precise : Helmholtz<br />
gave the value 1/a for what in Neumann was an indetermined constant 217.<br />
This "demonstration" became very famous as an indication of the heuristic power<br />
of the principle 218. Instead the fourth case 219 discussed in this section of the<br />
Erhaltung , concerning the interactions between two currents, became famous for<br />
the opposite reason: the lack of correct deductions from the principle.<br />
Here Helmholtz simply extended the previous formulation: tension forces<br />
provided by the batteries of the two circuits are<br />
A 1 J 1 <strong>and</strong> A 2 J 2,<br />
living forces identified <strong>with</strong> the heat produced by the current in the two circuits<br />
are<br />
J 2<br />
1 W 1<br />
<strong>and</strong> J21<br />
W2 plus 1<br />
a J1J dV<br />
2 dt<br />
<strong>and</strong> 1/a J1J2 dV/dt is interpreted as the living force of a circuit under the effect of<br />
the current circulating in the other one<br />
His claim that the results so obtained were in agreement <strong>with</strong> Weber's does not<br />
imply that his result was correct; Helmholtz in fact dismissed two kinds of<br />
potentials that, this time, really exist : the mutual potential of the two currents<br />
(electrokinetic energy) <strong>and</strong> the potential of a current on itself (selfinduction) 220.<br />
The conflicting outcome of the two cases discussed point to a serious problem: to<br />
what extent <strong>Helmholtz's</strong> principle was a useful heuristic device? In fact the first,<br />
correct, deduction was in a way a reinterpretation of already existing knowledge.<br />
Helmholtz did not include terms that could have been reasonably included not to<br />
violate Neumann's law. What about, for instance, the tension forces given by the<br />
potential JV of the current on the magnet? Have they to be taken in account or<br />
not? From a theoretical point of view we could easily give a formulation of the<br />
217 Helmholtz Erhaltung P.65.<br />
218 In 1873 it was still quoted <strong>and</strong> praised in Maxwell's Treatise Maxwell, James<br />
Clerk. A Treatise on Electricity <strong>and</strong> Magnetism. 2 Vols. 1873; Thomson J.J. ed of 3rd edition<br />
1891; repr New York: Dover,1954; Pp.190-3;but J.J.Thomson remarked in a note to third<br />
edition that the law of induction cannot be deduced through the principle of conservation of<br />
energy alone: another equation is needed. Ibidem P.192.<br />
219 Helmholtz Erhaltung Pp.67-8.<br />
220 see Planck Princip P.47
principle taking this potential into account, in analogy <strong>with</strong> the case (previously<br />
discussed) of the coupling of two magnets. But if we admit that potential we<br />
would not be able to rededuce Neumann's induction law, a law that is empirically<br />
corroborated. 221. In the second case terms unknown in 1847, but that indeed exist,<br />
were not predicted.The usefulness of the specific application of the concepts of<br />
tension <strong>and</strong> living forces is also questionable 222: what is the rational for defining<br />
aAJdt as tension forces? The equation in fact for Helmholtz still holds if A = 0,<br />
thus <strong>with</strong>out tension forces. Moreover, what is the reason to call the product JdV<br />
"the living force acquired by the magnet"? <strong>Helmholtz's</strong> efforts show thus that it is<br />
impossible an a priori precise theoretical deduction of the energy equations:<br />
experimental knowledge is required; on the other side, <strong>with</strong>out a guiding<br />
principle it is difficult to find <strong>and</strong> to interpret this experimental knowledge;<br />
moreover the expression of the principle can offer a ground for comparison for<br />
theories experimentally equivalent.<br />
This basic principle-theory-experiment interplay present in the Erhaltung will be<br />
permanently connected <strong>with</strong> the subsequent developments of the energy<br />
conservation theory. It is always difficult to take into account the different forms<br />
of energy involved in a specific process, <strong>and</strong> impossible to find the ultimate<br />
result 223.<br />
At the end of the section Helmholtz asserted that for some more complicated<br />
cases discussed by Neumann <strong>and</strong> Weber the principle cannot give precise<br />
determinations, but only qualitative indications, also because of the lack of<br />
experimental results 224.<br />
8 Conclusion<br />
Pp.111 <strong>and</strong> 114.<br />
221 See Planck Princip Pp.45-6.<br />
222 It was in fact discussed in Helm Energetik Pp.45-7.<br />
223 On the impossibility of a "primary" expression of energy see: Planck Princip<br />
224 The electromagnetic equations will be revised <strong>and</strong> rediscussed in 1854 during the<br />
controversy <strong>with</strong> Clausius.
The short <strong>and</strong> concise conclusion of the Erhaltung is mainly dedicated to<br />
physiological problems, <strong>Helmholtz's</strong> "professional" domain. Here again, together<br />
<strong>with</strong> a summary <strong>and</strong> reassessment of results previously expressed in the "Warme"<br />
<strong>and</strong> in the "Bericht", we find interesting new remarks.<br />
The main problem, as usual, was the formulation of the force equivalents of the<br />
energy balance. Helmholtz asserted that in the case of the vegetable world, any<br />
precise application of the principle is impossible for the lack of the necessary<br />
information. It can only be predicted that the tension forces stored are of<br />
chemical origin <strong>and</strong> that the only living forces absorbed are those of the<br />
"chemical solar rays" 225.<br />
The application of the principle to the animal world can be made <strong>with</strong> greater<br />
accuracy. Here Helmholtz, summarising the results of his previous research for<br />
the benefit of physicists, introduced for the first time the concepts of tension <strong>and</strong><br />
living forces in physiology: animals utilise a certain quantity of chemical tension<br />
forces <strong>and</strong> generate heat <strong>and</strong> mechanical forces. But Helmholtz believed that the<br />
mechanical work done by the animals is only a small quantity compared to the<br />
heat produced <strong>and</strong> could thus be omitted in the equation of the force equivalents.<br />
The corroboration of the conservation principle was related here to the following<br />
question: whether the combustion <strong>and</strong> conversion of the nutritive substances<br />
generate a quantity of heat equivalent to that produced by the animals. Helmholtz<br />
did not go into details here 226, but, on the basis of the Dulong <strong>and</strong> Despretz<br />
experiments, believed he could offer a positive answer, at least approximately.<br />
In my view this approach again shows that <strong>Helmholtz's</strong> main aim was to outline<br />
a general framework for the largest possible class of phenomena, independently<br />
of the actual experimental determination of the mechanical equivalent of heat <strong>and</strong><br />
thus independently of precise experimental corroboration. The exact value of the<br />
mechanical equivalent was not known, but not too relevant in this context: in<br />
asserting that the work done by the animals is a small percentage of the heat<br />
produced Helmholtz showed he was satisfied <strong>with</strong> a gross figure 227.<br />
Next comes a criticism of Carlo Matteucci's 1847 rebuttal of the conservation<br />
approach. Helmholtz showed that it was based on theoretical misunderst<strong>and</strong>ings<br />
<strong>and</strong> experimental inaccuracies 228.<br />
225 Helmholtz Erhaltung P.69.<br />
226 Helmholtz Erhaltung P.70<br />
227 Helmholtz Erhaltung P.70. See Kremer "Therm of Life" P.251.<br />
228 Helmholtz Erhaltung P.70-1.
Finally we get to the real conclusions, which are worth quoting at length.<br />
Helmholtz in fact did not claim to have demonstrated the principle, but only<br />
believed he had shown that the principle of conservation was "not in<br />
contradiction <strong>with</strong> any known fact in natural science but that it is corroborated in<br />
a remarkable way by a great number of such facts" 229.<br />
The sophisticated plan outlined at the beginning of the Einleitung has been<br />
carried out: <strong>with</strong> the greatest possible completeness the consequences of uniting<br />
the principle <strong>with</strong> the known laws of natural phenomena have been outlined. But<br />
Helmholtz was aware that his own extraordinary theoretical efforts lacked<br />
experimental corroboration: he did not claim to have achieved the latter, but<br />
instead that " the consequences outlined have still to wait for experimental<br />
confirmation". His goal was "to show to the physicists the theoretical, practical<br />
<strong>and</strong> heuristic relevance of the principle of conservation of force, whose<br />
exhaustive corroboration must be considered, probably, as one of the main tasks<br />
of physics in the near future" 230.<br />
It would certainly be difficult to find in scientific literature a paper expressing a<br />
clearer consciousness of the goals attempted <strong>and</strong> the results achieved. Both from<br />
the physical <strong>and</strong> the methodological point of view, the Erhaltung is a<br />
masterpiece.<br />
Questioning one root: the central force controversy <strong>with</strong><br />
Clausius (1852-54)<br />
The third stage of <strong>Helmholtz's</strong> struggle <strong>with</strong> 'energy' is characterised by<br />
a strong controversy he had <strong>with</strong> Clausius. The Erhaltung had not enjoyed great<br />
success. Helmholtz was now a physiologist in Koenigsberg <strong>and</strong> was trying to<br />
convince F.Neumann of his approach to the conservation principle. In the debate<br />
<strong>with</strong> Clausius, Helmholtz attempted to defend his own views <strong>and</strong> at the same<br />
time to qualify as an expert on physico-mathematical grounds, not a minor task<br />
for a physiologist. Clausius' criticisms started in 1852 <strong>and</strong> were extremely serious<br />
: he disclaimed, among others, not the physical probability but the mathematical<br />
necessity of central forces to fulfil vis viva conservation. In his answer,<br />
Helmholtz showed himself to be a great master of mathematical physics <strong>and</strong> of<br />
scientific methodology; through the introduction of a few (partly ad hoc) physical<br />
229 Helmholtz Erhaltung P.72.<br />
230 Helmholtz Erhaltung P.72.
hypotheses he managed not to lose ground. But <strong>Helmholtz's</strong> external success<br />
could only mask what in fact was a very serious internal problem in his research<br />
program. Clausius' criticism was explicitly recognized as sound only thirty years<br />
later, but Helmholtz must have realized its soundness immediately. In fact there is<br />
no further trace of central forces <strong>and</strong> of the dichotomy of potential <strong>and</strong> kinetic<br />
energies in the fourth stage of his commitment to the energy problems: the<br />
popularization of the doctrine.<br />
-----------------<br />
The Erhaltung did not raise enthusiastic reactions, apart from amongst the<br />
members of the Physikalische Gesellschaft. Despite Du Bois' pressure, Magnus'<br />
presentation of the work to Poggendorff was very cool. Magnus objected to what<br />
he supposed was an attempt at unifying physics through mathematics.<br />
Poggendorff refused to publish the paper in his Annalen because there were no<br />
new experimental results <strong>and</strong> it was too long 231. I believe the acceptance would<br />
have been different if Helmholtz had not mistranslated Joule's numerical results<br />
<strong>and</strong> if he had presented Joule's experiences as corroborating his own views.<br />
The essay, together <strong>with</strong> the philosophical introduction dropped for the<br />
presentation at the Physikalische Gesellschaft, was published by Reimer in<br />
Berlin, <strong>and</strong> Helmholtz, to his surprise, received an honorarium 232.<br />
Helmholtz after the Erhaltung returned to physiology <strong>and</strong> received the chair of<br />
physiology at Koenigsberg in 1849 233. But of course he did not give up his<br />
interest in energy conservation: in Koenigsberg he struggled to convince<br />
F.Neumann, who had already used an extended interpretation of the vis viva<br />
231 Koenigsberger H v H p.38; C.Jungnickel, R.McCormmach, Intellectual Mastery :<br />
vol.1, p.157; on the generational differences among German physicists see: Caneva, K."From<br />
Galvanism to Electrodynamics: The Transformation of German Physics <strong>and</strong> its Social<br />
Context." HSPS 9 (1978) 63-159.<br />
232 Koenigsberger H v H p.42.<br />
233 Extraordinary Professor <strong>and</strong> Director of the Physiological Institute: he became<br />
Professor in 1851.
conservation 234, on the theory of "force" conservation 235. Helmholtz worked <strong>with</strong><br />
Neumann's help on a paper to appear in the 1851 Annalen 236.<br />
In 1851 Helmholtz travelled around Germany (that is, German speaking states<br />
<strong>and</strong> towns) <strong>and</strong> met (together <strong>with</strong> physiologists) the most important physicists.<br />
In a letter he remarks that W.Weber received him <strong>with</strong> "less apparent cordiality<br />
than his brother in Leipzig" 237. For certain the feelings between the two were<br />
influenced by the deep contrast, plainly evident to both, between Weber's force<br />
law of 1846 238 <strong>and</strong> Weber's potential law of 1848 239 <strong>with</strong> <strong>Helmholtz's</strong> 1847<br />
approach.<br />
In 1852 Helmholtz added to his own mathematical knowledge, at least in part<br />
through F.Neumann, Green's theorems 240, which were to turn useful in the<br />
controversy that was about to start against Rudolf Clausius. In that very year, in<br />
fact, Clausius made some criticisms to the Erhaltung, criticisms that were based<br />
on potential theory <strong>and</strong> directed at the core of <strong>Helmholtz's</strong> program, that is to "the<br />
main advance on the investigations of Robert Mayer" 241.<br />
234 See <strong>Helmholtz's</strong> references already in the 1847 "Bericht".<br />
235 Koenigsberger H v H p.64.<br />
236 Helmholtz, Hermann. "Ueber die Dauer un den Verlauf der durch<br />
Stromesschwankungen inducirten elektrischen Ströme." In Poggend Ann 83 (1851): 505-40.<br />
See: Jungnickel,C. <strong>and</strong> McCormmach, R. Intellectual Mastery : vol.1, p. 162.<br />
237 Koenigsberger H v H p. 81.<br />
238 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen über ein allgemeines<br />
Grundgesetz der Elektrischen Wirkung." In Werke 6 vols. Berlin, 1892-4. Vol.3, pp.25-214.<br />
See: Koenigsberger H v H p.100.<br />
239 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen" In Pogg Ann 73<br />
(1848): 193. Tr. in Taylor's Scientific Memoirs 5 (1852): 489-529. W.Weber already in 1848<br />
had shown that his force law of 1846 could be derived, contra <strong>Helmholtz's</strong> Erhaltung, from a<br />
potential. This implied that the work done by the force was a total differential <strong>and</strong> that the<br />
force law did not violate the impossibility of perpetual motion. But, given that the force law<br />
included velocities <strong>and</strong> accelerations, the potential included a kinetic term. Actually the kinetic<br />
<strong>and</strong> positional part of the potential could not be sharply divided, a serious blow against<br />
<strong>Helmholtz's</strong> approach. Weber's law soon acquired a leader position in electrodynamics, but his<br />
potential was not accepted as a viable interpretation of energy conservation till the early<br />
seventies.<br />
240 Koenigsberger H v H p.100.<br />
241 Koenigsberger H v H p.118.
This discussion was to outline a third, more mathematical, formulation of the<br />
principle of conservation of energy, after the correlation of forces (conversion<br />
<strong>with</strong> constant coefficients) <strong>and</strong> conservation of "force" (variation of vis viva<br />
equals the sum of tension forces).<br />
The problem of appreciating the specific characteristics of <strong>Helmholtz's</strong> innovative<br />
theoretical approach was a serious one: Helmholtz claimed recognition among the<br />
physicists, both experimental <strong>and</strong> mathematical. But to receive it he had to fight a<br />
hard battle. Only J.J. Jacobi, the mathematician, encouraged him, recognising the<br />
links between the Erhaltung <strong>and</strong> the works of the "great mathematicians of the<br />
preceding century", <strong>and</strong> despite the contrary opinion of Lejeune-Dirichlet <strong>and</strong><br />
Eisenstein 242. On top of Magnus' <strong>and</strong> Poggendorff's criticisms, Karsten in 1850 in<br />
the Fortschritt der Physik im Jahre 1847 inserted the report on the Erhaltung<br />
in the physiological section, <strong>and</strong> only later, in 1855, commenting on the 1850-51<br />
debate on the "Theorie der Wärme", Helmholtz managed to give news of his<br />
paper in the physical section 243 of that journal.<br />
In this context Clausius' criticisms were received <strong>with</strong> great preoccupation 244,<br />
given that Clausius was already a well- known mathematical physicist, <strong>and</strong><br />
answered in detail.<br />
Rudolf Clausius <strong>and</strong> Hermann Helmholtz knew each other rather well: together<br />
<strong>with</strong> Wiedemann they regularly took their meals together in the winter of 1847 245<br />
<strong>and</strong> since 1848 had been in the habit for a long time of meeting almost daily 246.<br />
Clausius was in fact a participant in Magnus' colloquium from the beginning. In<br />
1850 he published a famous paper on the mechanical theory of heat, in which he<br />
defined the two fundamental principles of thermodynamics. He derived a value<br />
242 Helmholtz 1882 final appendix to the Erhaltung : WA 1 p.71; Koenigsberger H v<br />
H p. 43; for a modern denial of the physico-mathematical relevance of theErhaltung see:<br />
Truesdell, Clifford. The Tragicomic History of Thermodynamics. 1822-1854. New York:<br />
Springer,1980. Pp.161.<br />
243 See also: Heimann "Helmholtz <strong>and</strong> Kant" p.233 n.104. Helmholtz in 1855 started<br />
publishing a series of six "Berichts" in the Fortschritte der Physik (1855-9) that can now be<br />
seen as the first history of energy conservation.<br />
244 Koenigsberger stresses <strong>Helmholtz's</strong> preoccupation, so great to make him feel<br />
"betrayed" by a former friend. Koenigsberger H v H p. Actually Clausius' initial criticisms<br />
were very light <strong>and</strong> mixed <strong>with</strong> praise.<br />
245 Jungnickel,C. <strong>and</strong> McCormmach, R. Intellectual Mastery : vol.1, p.256.<br />
246 Koenigsberger H v H p.115.
for the mechanical equivalent of heat <strong>and</strong> in 1852 in his first paper on<br />
electricity 247, which is at the origin of the controversy, he "applied his heat laws<br />
to electrical discharge <strong>and</strong> thermoelectricity" 248. He thus united mechanics, heat<br />
<strong>and</strong> electricity in a thorough study based on mechanical principles. Clausius later<br />
dedicated great efforts to an electrodynamical theory that he saw as part of his<br />
own mechanical theory of heat 249. In fact in 1865 he started republishing his<br />
papers on electrodynamics in the second volume of the first edition of Die<br />
Mechanische Wärmetheorie 250. In 1879 he went much further: for the second<br />
edition of the book he rewrote the entire second volume <strong>and</strong> dedicated it entirely<br />
to his own interpretation of electrodynamics; thus the book appeared <strong>with</strong> the<br />
surprising title of Die Mechanische Wärmetheorie Bd2 Elektrodynamik : Die<br />
Mechanische Beh<strong>and</strong>ung der Elektricität 251. In this volume, where Clausius<br />
"demonstrated" his electrodynamic force law of 1875 252 based on a kinetic<br />
potential, very little room is left for Maxwell's <strong>and</strong> <strong>Helmholtz's</strong> electrical theories.<br />
247 Clausius, Rudolf. "On the Mechanical Equivalent of an Electric Discharge, <strong>and</strong> the<br />
Heating of the Conducting Wire which accompanies it." In Tyndall <strong>and</strong> Francis Scientific<br />
Memoirs on Natural Philosophy 1 (1853): 1-32 <strong>and</strong> 200-9.<br />
activity.<br />
248 Jungnickel,C. <strong>and</strong> McCormmach, R. Intellectual Mastery vol.1, p.167.<br />
249 Historians have payd very little attention to this not minor aspect of his scientific<br />
250 Clausius, Rudolf. Die Mechanische Wärmetheorie 2 vols Braunschweig, 1865.<br />
The first volume was translated in English in 1867.<br />
251 The first volume had already appeared: Clausius, Rudolf. Die Mechanische<br />
Wärmetheorie 2nd ed 1st vol. Braunschweig, 1876. (An English translation of the second<br />
edition of the first volume appeared in 1879: The Mechanical Theory of Heat London 1879).<br />
The second volume followed three years later: Clausius, Rudolf. Die Mechanische<br />
Wärmetheorie 2nd ed. 2nd vol. Braunschweig,1879. A French translation of both volumes of<br />
the third German edition was made by F.Folie <strong>and</strong> E.Ronkar, Bruxelles :1897-8.<br />
252 Clausius' law admitted, as Weber's one, forces depending on velocities <strong>and</strong><br />
accelerations; but, at variance <strong>with</strong> Weber, the velocities were 'absolute' (respect to the ether)<br />
<strong>and</strong> thus the forces were also non central. Clausius' ether explains <strong>Helmholtz's</strong> 1882 remark<br />
about the loss of intellegibility . See n.94 above. Clausius, Rudolf. "Ueber ein neues<br />
Grundgesetz der Elektrodynamik." In Pogg Ann 156 (1875): 657-60; tr. in Phil Mag 1 (1876):<br />
69-71.
In 1876 Clausius, discussing 253 his own electrodynamic law of the previous year,<br />
clarified his interpretation of the conservation principle, in a way that is strongly<br />
connected <strong>with</strong> the approach of his early works on electricity: the only condition<br />
needed to fulfil the principle of conservation is that the work done by the forces<br />
is a total differential. No physical interpretation of potential energy or of the<br />
energy concept is given, conservation is still defined along the lines of the<br />
analytical tradition as vis viva conservation. Clausius' approach derives from his<br />
committment to mathematical potential theory 254 <strong>and</strong> to his identification of work<br />
<strong>with</strong> the difference of potential.<br />
All this shows that Clausius' 1852 remarks on <strong>Helmholtz's</strong> results were by no<br />
means casual, but originated <strong>and</strong> deeply rooted in Clausius' research<br />
programme.<br />
A main difference in the interpretation of the energy concept was to create a<br />
lifelong barrier between these two champions of the mechanical view of nature:<br />
Helmholtz believed in two forms of energy sharply divided <strong>and</strong> based on central<br />
forces depending only on distance, while Clausius admitted forces that were not<br />
central <strong>and</strong> depended also on velocities <strong>and</strong> accelerations as far as they admitted<br />
a potential; the fact that in this (kinetic) potential appeared terms including both<br />
positions <strong>and</strong> velocities was not seen as a problem by Clausius . The existence<br />
itself of a potential was considered as the fulfilment of the impossibility of<br />
perpetual motion.<br />
The whole controversy of 1852-54 took place in the pages of the Annalen der<br />
Physik 255, a clear indication that both Clausius' <strong>and</strong> <strong>Helmholtz's</strong> theoretical<br />
contributions were now fully accepted by Poggendorff 256.<br />
253 Clausius, Rudolf. "Ueber das Verhalten des elektrodynamischen Grundgesetzes<br />
zum Prinzip von der Erhaltung der Energie und über eine noch weitere Vereinfachung des<br />
ersteren." In Pogg Ann 157 (1876):489-94; tr.in Phil Mag s5 1 (1876): 218-21.<br />
254 Clausius published many editions of his textbook on the Potential <strong>and</strong> the Potential<br />
Function starting from 1859. A French translation of the second German edition of 1866<br />
appeared in 1870: Clausius, Rudolf. De la fonction potentielle et du potentiel. Tr by F.Folie.<br />
Paris:Gauthier-Villars, 1870.<br />
255 Comments on this controversy are in Planck Prinzip Pp.48-50, Koenigsberger H<br />
v H Pp.115-20, Heimann "Helmholtz <strong>and</strong> Kant" Pp.234-7, Jungnickel,C. <strong>and</strong> McCormmach,<br />
R. Intellectual Mastery p.163.<br />
256 On the cultural policy of the journal see: Jungnickel,C. <strong>and</strong> McCormmach, R.<br />
Intellectual Mastery Vol.1, Pp. 113-128.
The controversy started <strong>with</strong> some remarks on <strong>Helmholtz's</strong> Erhaltung in the<br />
notes of Clausius' 1852 paper on electric discharge. Helmholtz answered in a<br />
note of his 1853 257 paper on the distribution of electrical currents in material<br />
conductors. In the same year 258 Clausius fought back <strong>and</strong> in 1854 259 Helmholtz<br />
expressed his views at length , in a paper to be reprinted in the energy section in<br />
the first volume of his collected works. A short comment of Clausius in 1854 260<br />
closed the debate. As already seen 261, later on in 1882, in the appendix to the<br />
reprint of the Erhaltung in the same first volume of the collected works,<br />
Helmholtz admitted, in part <strong>and</strong> <strong>with</strong>out quoting Clausius, that some of his old<br />
objections were correct.<br />
There were three main points of debate <strong>and</strong> I will deal <strong>with</strong> them separately: a)<br />
the definition of potential; b) the conceptual model of central forces; c)<br />
accusations of lack of precision.<br />
a) Helmholtz had introduced the concept of Spannkraft (potential energy) <strong>with</strong>out<br />
going into a discussion of the concept of potential. The peculiarity of <strong>Helmholtz's</strong><br />
approach is that he jumps from the vis viva theorem to the conservation of<br />
"force" principle, <strong>with</strong>out following the now st<strong>and</strong>ard pattern of f x ds = work,<br />
work as total differential = difference of potential. In the case of central<br />
Newtonian forces work is indeed a total differential <strong>and</strong> thus the two approaches<br />
intersect considerably. In fact, as discussed in the previous section, Helmholtz<br />
introduces in a straightforward way in the second chapter of the Erhaltung the<br />
sum of tension forces (potential energy) <strong>and</strong> the energy concept (the variation of<br />
vis viva equals the variation of the sum of tension forces, the sum of vis viva <strong>and</strong><br />
tension forces is a constant) as conceptual <strong>and</strong> physical entities. Instead he<br />
introduces the term potential only in the fifth chapter, <strong>with</strong> a physical grasp not<br />
completely refined <strong>and</strong> <strong>with</strong>out knowledge of Green's contributions, but "in<br />
257 Helmholtz, Hermann. "Ueber einige Gesetze der Vertheilung elektrischer Ströme<br />
in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche." In Pogg Ann<br />
89 (1853): 211-33, 352-77. Repr. in WA 1 pp.475-519; n.1 p.488.<br />
258 Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber die<br />
Erhaltung der Kraft." In Pogg Ann 89 (1853): 568-579.<br />
259 Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In<br />
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, pp.76-93.<br />
260 Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber die<br />
Erhaltung der Kraft", zweite Notiz." In Pogg Ann 91 (1854): 601-604.<br />
261 See nn.125, 126, 127 above. Helmholtz WA 1 p.70.
conformity <strong>with</strong> Gauss in his magnetical researches" 262 as the sum of tensions<br />
consumed by the motion from infinity to r, <strong>and</strong>, equivalently, as the sum of the vis<br />
viva produced. Thus :<br />
"the increase of vis viva in any movement must be considered equal to the<br />
difference of the potential at the end of the trajectory <strong>with</strong> respect to the potential<br />
at the beginning" 263.<br />
That is, the sum of tension forces is equivalent to the difference of potentials:<br />
− ∫ r<br />
R<br />
φ dr = e1e2 R − e1e2 r<br />
<strong>and</strong>, of course, also to the gain in vis viva caused by passing from the distance R<br />
to r.<br />
Helmholtz then introduces the concept of the potential of a body in itself <strong>and</strong> of<br />
the potential of a body on another. In a specific case given as an example, he<br />
calculates the difference of potentials before <strong>and</strong> after the movement of a<br />
quantity of electricity. This difference is defined as equivalent to the quantity of<br />
work done:<br />
-(V + (Wa +Wb)/2) 264<br />
As already seen Clausius in 1852 criticised Helmholtz assertion that in the above<br />
expression the potential W of a body on itself is not equal to the corresponding<br />
work done, but twice as much (the corresponding work is in fact W/2). The<br />
accusation of not having understood the deep relations between potential <strong>and</strong><br />
work was a serious one from Clausius' point of view.<br />
Clausius in the same1852 paper had established a relation between vis viva <strong>and</strong><br />
potential different from that of <strong>Helmholtz's</strong> Erhaltung. He started from the vis<br />
viva theorem <strong>and</strong> equated the increase of vis viva to the quantity of mechanical<br />
work produced during the same time in the system 265. Clausius did not accept<br />
<strong>Helmholtz's</strong> "sum of tension forces" (potential energy) <strong>and</strong> the corresponding<br />
interpretation of the conservation principle. For Clausius, work, being in most<br />
cases a total differential <strong>and</strong> thus its integral depending only on the initial <strong>and</strong><br />
final positions, can be identified <strong>with</strong> a difference of potentials. The potential of<br />
an exterior system of masses on a given system is introduced as the function:<br />
262 Helmholtz Erhaltung p. 38.<br />
263 Helmholtz Erhaltung p. 38.<br />
264 Helmholtz Erhaltung Pp.42-3.<br />
265 Clausius "Electric Discharge" p.3.
where μ are immovable masses, m the masses of the system <strong>and</strong> ρ the distance of<br />
the two masses from each other.<br />
In a similar way:<br />
is the potential of the given system of masses upon itself.<br />
"The work consists simply in the increase of these potentials" 266.<br />
Thus in 1852 Clausius explicitly asserted that the potential is work stored in the<br />
system. Work as total differential <strong>and</strong> difference of potential are identical<br />
concepts. This is a statement of the greatest relevance, but very different from<br />
<strong>Helmholtz's</strong>. In the Gauss-Clausius tradition energy would never become a<br />
physical quantity. The principle of conservation was often to be called, following<br />
the old tradition, the vis viva conservation 267 <strong>and</strong> the only really important<br />
requirement was that work be a total differential. This interpretation left open the<br />
possibility that forces other than the central Newtonian ones could satisfy the<br />
conservation principle, if the work done by these forces satisfies the mentioned<br />
requirement. An answer to Clausius was thus due.<br />
In 1853 Helmholtz clarifies his own use of the term "free tension" in the fifth<br />
paragraph of the Erhaltung asserting that its definition "is identical to what<br />
Gauss defined as potential <strong>and</strong> Green as potential function" 268. Thus in 1853<br />
Helmholtz explicitly acknowledges a result of the unifying power of the<br />
mathematical potential theory: the concept of electrical tension, formerly <strong>with</strong><br />
Volta <strong>and</strong> Ohm density of the elastic fluid called electricity, had been redefined<br />
by Kirchhoff in 1849, both for static electricity <strong>and</strong> for currents, as difference of<br />
electrical potential 269. Electrostatics <strong>and</strong> galvanism had thus been unified.<br />
In a note in the same page 270, Helmholtz replied to Clausius' criticisms of 1852<br />
asserting that the problem of the definition of the potential does not imply any<br />
266 Clausius "Electric Discharge" in the note of p.5, quotes both Gauss <strong>and</strong> Green.<br />
267 See for instance: Riemann Schwere Elektricität Magnetismus ; Sturm, M. Cours<br />
de Mécanique , M.Prohuet ed. Vols. 2. Paris: Mallet-Bachelier, 1861.<br />
268 Helmholtz "Einige Gesetz" p.224.<br />
269 Kirchhoff, Gustav. "On a Deduction of Ohm's Law, in Connection <strong>with</strong> the<br />
Theory of Electrostatics" Phil Mag s3 37 (1850): 463-8; translated from Pogg Ann 78<br />
(1849) p.506.<br />
270 Helmholtz "Einige Gesetz" p.224.
substantial disagreement : <strong>Helmholtz's</strong> own definition of potential can be<br />
considered correct <strong>and</strong> his own development consistent, even if he admits that it<br />
was not a good definition, because it was not in agreement <strong>with</strong> F.Neumann's.<br />
Helmholtz had simply assumed for the potential of a body (system of bodies) on<br />
itself double the value than had Clausius. That is, he had considered for the<br />
potential of the couple A-B both the work done to transport A to its position<br />
under the force of B <strong>and</strong> viceversa to transport B under the force of A. But the<br />
work done in the example under discussion had been calculated correctly.<br />
Clausius could not be satisfied <strong>with</strong> this answer: He spends six pages in 1853 271<br />
clarifying that <strong>Helmholtz's</strong> approach rests on what from his point of view was a<br />
serious mistake, the conceptual difference between potential <strong>and</strong> work. For<br />
Clausius the correct result for the work done in the example at issue is only due<br />
to the compensating effect of two opposite mistakes. He first shows that the great<br />
importance of the concept of potential of two masses, each one respect to the<br />
other, is connected <strong>with</strong> its equivalence "in a specific but frequent case, <strong>with</strong> that<br />
of mechanical work" 272. Work has to be considered as one of the most important<br />
quantities of the whole of mechanics <strong>and</strong> mathematical physics, "<strong>and</strong> also<br />
Helmholtz in his essay utilized it <strong>with</strong> this meaning" 273.<br />
Thus if we have to introduce the concept of "potential of a mass on itself" <strong>with</strong><br />
twice the value of the corresponding mechanical work, priority should be given<br />
to the other concept of potential (that is, to the potential of a body on another<br />
which is equivalent to the work). The opposite choice of Helmholtz is a flaw" 274.<br />
But a more serious mistake, for Clausius, is to have added together in the same<br />
expression a potential corresponding to the work (the potential of a body on<br />
another) <strong>and</strong> one corresponding to twice the work (the potential of a body on<br />
itself). In the simple example given by Helmholtz the final result is correct only<br />
because the mistake of the double value of the potential of a mass on itself is<br />
compensated by the particular choice of the values of the electricities involved.<br />
<strong>Helmholtz's</strong> answer in 1854 on this point is interesting: he recalls that in that<br />
particular example his definition was supposed to hold only under specific<br />
hypotheses, that in general his definition of potential <strong>and</strong> work are correct, for<br />
example for electrified bodies <strong>and</strong> for magnets, <strong>and</strong> asserts that in the case of the<br />
271 Clausius "Einige Stellen" Pp.568-74.<br />
272 Clausius "Einige Stellen" p.569.<br />
273 Clausius "Einige Stellen" p.569.<br />
274 Clausius "Einige Stellen" Pp.569-70.
demonstration criticised by Clausius, a wrong work equivalent for the potential<br />
was not presupposed, on the contrary the very goal of the demonstration was to<br />
find the work equivalent 275.<br />
In no way could the different methodologies be better clarified. In a short answer<br />
in 1854 Clausius reasserts his point 276 <strong>and</strong> finally in 1882 in the last appendix to<br />
the Erhaltung, Helmholtz accepts the criticism that his definition of the potential<br />
of a charge (body) on itself is less convenient than the one that equates it <strong>with</strong> the<br />
work done. Still he denies giving a wrong value for the work in 1847: it was<br />
asserted in the Erhaltung that the work was half the mentioned potential.<br />
To summarise this point of the controversy: for both Helmholtz <strong>and</strong> Clausius the<br />
difference of potential has to be related to the work done; Helmholtz in the<br />
Erhaltung faced some difficulties in the definition, partly for his ignorance of<br />
Green's papers (he in fact only quoted Gauss) <strong>and</strong> partly for his physical rather<br />
than mathematical grasp of the problem. Nonetheless it is remarkable that his<br />
"free tension" was later to be identified <strong>with</strong> the "potential function" <strong>and</strong> that, as<br />
everywhere else in the Erhaltung , he started from the theoretical assumption of<br />
the equality between quantity of Spannkraft <strong>and</strong> variation of vis viva to find the<br />
work done in the specific cases. The equivalence between potential <strong>and</strong> work is<br />
not, as in Clausius, a prerequisite, but a consequence of the existence of a<br />
potential energy. In the above quoted case of the electric discharge (i.e. where a<br />
variation of the charge distribution exists), Helmholtz knows that the tension<br />
forces would then be modified during the discharge, <strong>and</strong> thus builds up his<br />
example to avoid the problem., through considerations of symmetry. Without<br />
doubt Clausius was right in pointing out that <strong>Helmholtz's</strong> definition of potential of<br />
a charge on itself was twice the work done, but in the context of <strong>Helmholtz's</strong><br />
research this was a minor mistake <strong>and</strong> does not imply that Helmholtz had not<br />
understood, despite his lack of knowledge of part of the literature, the relations<br />
between work <strong>and</strong> potential. Moreover Clausius had an advantage: after<br />
<strong>Helmholtz's</strong> 1847 essay <strong>and</strong> before Clausius 1852 paper the relations between<br />
potential <strong>and</strong> work had been clarified by Weber (1848) <strong>and</strong> Kirchhoff (1849).<br />
b) The second point of the controversy is a much more serious problem for<br />
Helmholtz: Clausius challenged the necessity of the central force hypothesis, one<br />
of the roots of <strong>Helmholtz's</strong> definition of energy.<br />
275 Helmholtz "Erwiderung" Pp.76-7.<br />
276 Clausius "Zweite Notiz" Pp.601-2.
Clausius' approach is in fact different: starting from the same work-vis viva<br />
equation as Helmholtz does, he recognizes that in certain cases the work done is<br />
a complete differential. Thus the vis viva can be equated to a difference of<br />
potential. But Clausius does not mention at this stage the potential energy, nor<br />
does he think that it should be a positional term. Even much later, the principle of<br />
conservation of energy for Clausius was meant to be the requirement that the<br />
work be expressed as a total differential. Clausius believed in the kinetic theory<br />
of heat, but he did not apply to it the dichotomy of vis viva <strong>and</strong> tension forces as<br />
Helmholtz had done in the fourth chapter of the Erhaltung . Clausius asserted in<br />
1852 that<br />
"heat consists in a motion of the ultimate particles of bodies <strong>and</strong> is a measure of<br />
the vis viva of this motion" 277.<br />
Thus the work done, or the difference of potential between the initial <strong>and</strong> final<br />
conditions, includes this effect too:<br />
" The sum of all the effects produced by an electric discharge is equal to the<br />
increase of the potential of the entire electricity upon itself" 278.<br />
Mechanical work, electricity <strong>and</strong> heat are unified here but "potential energy" <strong>and</strong><br />
"energy" do not appear. Even farther away from Clausius' perspective is the idea<br />
of a conservation of the sum of sharply separated kinetic <strong>and</strong> positional terms. In<br />
the paper mentioned, Clausius too relies on the model of central Newtonian<br />
forces depending only on distance, but introduces it only as a case which is<br />
mathematicallly simple, often occurring in physical situations:<br />
"The determination of the work may be much simplified in particular cases which<br />
very often present themselves" 279.<br />
Clausius does not believe central Newtonian forces to be a conceptual model<br />
necessary for the intelligibility of nature. This criticism of <strong>Helmholtz's</strong> approach<br />
does not appear in the 1852 paper, but after <strong>Helmholtz's</strong> remarks of 1853,<br />
Clausius in the same year expressed his views on this point rather clearly.<br />
The central issue is immediately faced, "not for personal reasons, but because it<br />
is connected <strong>with</strong> problems of general scientific interest" 280. The problem is<br />
clearly defined: Helmholtz asserted that the vis viva principle holds only when<br />
the acting forces can be decomposed in forces acting on material points, in the<br />
277 Clausius "Electrical Discharge" p.5<br />
278 Clausius "Electrical Discharge" p.6<br />
279 Clausius "Electrical Discharge" p.4.<br />
280 Clausius "Einige Stellen" p.574.
directions that join them <strong>and</strong> whose intensity depends only on the distance, that is<br />
only for central forces. But Clausius claims that the demonstration given by<br />
Helmholtz does not justify the assertion.<br />
In fact, during the demonstration, one of the very statements to be derived, that<br />
the intensity of the force is a function of the distance, is presupposed 281. This<br />
presupposition is part of <strong>Helmholtz's</strong> model as expressed in the Einleitung. Thus<br />
the principle of vis viva is either unnecessary to derive the central forces (if we<br />
already presuppose that the intensity is a function of the distance) or is<br />
insufficient (if we do not make that assumption).<br />
Moreover, Clausius asserted that from the vis viva theorem, assuming that the<br />
vis viva is a function only of the space coordinates, we can derive that the work<br />
too is a total differential <strong>and</strong> also that the force acting is a function of the space<br />
coordinates. But this does not imply that the force be central. The only possible<br />
implication for Clausius is that, if one of the two conditions hold (direction or<br />
intensity) the other does too.<br />
Finally Clausius, while not questioning the "physical probability", denies the<br />
"mathematical necessity" of central forces 282.<br />
Helmholtz, apart from deriving them from the vis viva principle in the second<br />
chapter, had assumed in the Introduction that a point acts in different directions<br />
<strong>with</strong> the same force. Clausius does not consider this as evident: the contrary is<br />
not "unthinkable". Moreover in <strong>Helmholtz's</strong> first chapter of the Erhaltung, when<br />
the principle of virtual velocities is derived from the vis viva principle it is also<br />
asserted that the result is that the forces of two points act on the direction joining<br />
them. But for Clausius this, again, was part of the assumptions <strong>and</strong> cannot be<br />
considered a deduction. <strong>Helmholtz's</strong> whole attempt to demonstrate the<br />
mathematical need for central forces is, for Clausius, a flaw.<br />
The core of <strong>Helmholtz's</strong> program was seriously shaken. No doubt he was deeply<br />
worried by the criticisms for he dedicated a great effort to preparing a twenty<br />
page paper, to be published in 1854 <strong>and</strong> reprinted in 1882 in the energy section<br />
of the WA 1, after the Erhaltung.<br />
Replying to Clausius in this paper, Helmholtz asserted that he could derive from<br />
the vis viva principle the need for the forces to be central, <strong>with</strong>out further<br />
hypotheses, because he expressed the principle only <strong>with</strong> reference to the relative<br />
positions of the points <strong>and</strong> not to the absolute ones. In <strong>Helmholtz's</strong> view this<br />
281 Clausius "Einige Stellen" p.575.<br />
282 Clausius "Einige Stellen" p.577.
means taking into account the actual bodies <strong>and</strong> not "purely imaginary systems of<br />
coordinates" 283. Helmholtz admits that he assumed that the vis viva depends only<br />
on the positions of the points <strong>and</strong> thus only on their distance. If this version of the<br />
principle is accepted Helmholtz can derive, through a massive use of<br />
mathematics, both the hypotheses leading to central forces (the intensity depends<br />
on the distance <strong>and</strong> the direction is in the joining line) also for the case in which<br />
there is a point moving under the action of a material element. In this case, in<br />
principle, the vis viva of the point should depend on the orientation of the<br />
distance. Furthermore, Helmholtz explicitly asserts that the only implicit<br />
principle used, the superposition of the effects for the force, if accepted<br />
reinforces his conclusions 284.<br />
Clausius quick answer is very short, he denied that in the Erhaltung the concern<br />
was only <strong>with</strong> relative positions <strong>and</strong> reasserted his views on the impossibility of a<br />
mathematical deduction of the central forces 285.<br />
In 1882 once more Helmholtz had to accept the criticism, but <strong>with</strong> some<br />
qualifications: he now admits that the demonstration in the second chapter needs<br />
a restriction, that is: forces that depend also on velocities <strong>and</strong> accelerations are in<br />
agreement <strong>with</strong> the vis viva theorem, but not <strong>with</strong> the principle of action <strong>and</strong><br />
reaction 286. Nevertheless they are in agreement <strong>with</strong> the impossibility of perpetual<br />
motion. Helmholtz asserts in appendix 2 that this point was clarified for him by<br />
Lipschitz. He does not quote the controversy <strong>with</strong> Clausius, <strong>and</strong> in appendix 3<br />
while quoting Clausius' electrodynamic law does not quote Weber's.<br />
In a section below I will analyse the role of <strong>Helmholtz's</strong> expression of the energy<br />
concept <strong>and</strong> of the conservation principle in the electromagnetic debate of the<br />
70's. It is sufficient for the moment to point out that the root of the problem, the<br />
supposed necessity for energy conservation of central forces, was already explicit<br />
after Weber's 1848 expression of a kinetic potential <strong>and</strong> Clausius' 1853<br />
criticisms.<br />
c) Other minor points were part of the controversy: Clausius in 1852 287, while<br />
praising <strong>Helmholtz's</strong> Erhaltung, claims that some parts were incorrect; he<br />
283 Helmholtz "Erwiderung" p.82.<br />
284 Helmholtz "Erwiderung" Pp.88-9.<br />
285 Clausius "Zweite Notiz" p.604.<br />
286 Helmholtz WA 1. Appendix 2 <strong>and</strong> 3 to the Erhaltung; pp.68-70.<br />
287 Clausius "Electrical Discharge" p.6
quotes 288 <strong>and</strong> criticises a proposition of Vorsselman de Heer accepted in the<br />
Erhaltung ("the total heat which is provoked in the entire circuit by an electric<br />
discharge is independent of the nature of the circuit"). The same for an expression<br />
of Riess 289 referring to the independence of the heat of discharge from the nature<br />
of the connecting wires.<br />
Clausius in 1852 also criticises <strong>Helmholtz's</strong> interpretation of Holtzmann,<br />
asserting that the latter did not believe in the consumption of heat, but in its<br />
invariability. This is a point which Helmholtz seems to have misunderstood <strong>and</strong><br />
that seems to imply a serious flaw in his analysis of the work equivalent of heat in<br />
the fourth chapter of the Erhaltung. Clausius ends his 1853 paper <strong>with</strong> an<br />
acknowledgement <strong>and</strong> an appreciation of <strong>Helmholtz's</strong> achievements in the<br />
Erhaltung, despite the criticisms 290.<br />
Helmholtz in 1854 answers, in a detailed <strong>and</strong> balanced way, the specific points<br />
about de Heer <strong>and</strong> Riess <strong>and</strong> admits his own mistake as far as the interpretation<br />
of Holtzmann is concerned 291. He finally adds four new points to the<br />
electromagnetic research of the Erhaltung , on the basis of Poisson's magnetic<br />
induction, of his own researches of 1853 on the oscillations of induced currents.<br />
He found that a galvanic current has an electrodynamic potential in itself,<br />
proportional to the square of the current intensity. If the circuit is interrupted<br />
there is a conversion of this "force equivalent" in heat either in the spark or in the<br />
extracurrent 292. Finally he shows that F.Neumann's law of induction through<br />
magnets or currents is in agreement <strong>with</strong> conservation of energy.<br />
Clausius' final remarks acknowledge the clarifications <strong>and</strong> only insists on the<br />
problem of central forces 293.<br />
The result of the controversy is remarkable: if not the great success claimed by<br />
Koenigsberger 294, Helmholtz achieved some success. He had shown himself to be<br />
also a skilled mathematical physicist, able to debate <strong>with</strong> a researcher as clever<br />
as Clausius. In the circumstances this was a particularly difficult achievement for<br />
Helmholtz: Clausius was right (as later indirectly acknowledged by Helmholtz<br />
288 Clausius "Electrical Discharge" p.16<br />
289 Clausius "Electrical Discharge" p.21<br />
290 Clausius "Einige Stellen" Pp.578-9.<br />
291 Helmholtz "Erwiderung" p.90.<br />
292 Thus correcting his "wrong" deductions of 1847.<br />
293 Clausius "Zweite Notiz" p. 604.<br />
294 Koenigsberger H v H p.116.
himself) in the specific instances of a) the derivation of the central forces from the<br />
principle of vis viva <strong>and</strong> b) of the definition of potential .<br />
<strong>Helmholtz's</strong> strength was founded on his broad, general viewpoint <strong>and</strong> on the<br />
great heuristic <strong>and</strong> justificatory power of his approach, which allowed a<br />
systematic application to a variety of natural phenomena, <strong>and</strong> is not founded on<br />
the individual, specific demonstrations <strong>and</strong> applications. This was, <strong>with</strong> great<br />
chivalry, acknowledged by Clausius himself. The debate <strong>with</strong> Clausius being<br />
based on specific instances, Helmholtz was in a weak position ( Clausius was<br />
actually formally right in most criticisms) <strong>and</strong> it is remarkable that he managed<br />
not to let Clausius prevail. His target of being accepted among mathematical<br />
physicists was achieved. The 1853 paper was the third he published in the<br />
Annalen, <strong>and</strong> as already recalled, the whole controversy was published in that<br />
journal, which occasioned a greater diffusion in Germany of the Erhaltung 's<br />
ideas.<br />
<strong>Helmholtz's</strong> fame spread perhaps in Britain before at home. W.Thomson read the<br />
Erhaltung in 1852 295 <strong>and</strong> at DuBois-Reymond's suggestion Tyndall translated<br />
<strong>and</strong> published it in 1853, together <strong>with</strong> Clausius' 1852 paper. Helmholtz met<br />
Tyndall in 1853 in Berlin, which began a lifelong friendship, <strong>and</strong> when, in the<br />
same year, he participated in the meeting of the British Association for the<br />
Advancement of Science at Halle he discovered he was better known than in<br />
Germany 296.<br />
But the result of the controversy <strong>with</strong> Clausius was not <strong>with</strong>out a lesson for<br />
Helmholtz, as will be seen in the next section.<br />
Popular conservation of "force" : where have the central<br />
forces gone ? (1854-64)<br />
In a series of lectures, most of which are not included in the<br />
Wissenschaftliche Abh<strong>and</strong>lungen , Helmholtz presented to a large public the<br />
universal validity of the theory of conservation of energy. In so doing, he only<br />
utilized his very first approach to the problem: the conversion of forces through<br />
constant coefficients, the related extension of the impossibility of perpetual<br />
295 Planck Prinzip p.68.<br />
296 Koenigsberger H v H p.109 (Tyndall), p.113 <strong>and</strong> 120.
motion to all natural forces, the kinetic theory of heat <strong>and</strong> its mechanical<br />
equivalent. That is, he only utilized one of the two conceptual roots of the<br />
Erhaltung, <strong>and</strong> did not talk of the requirement that all the forces be central <strong>and</strong><br />
that the energies be reduced to only two kinds. Moreover, he ab<strong>and</strong>oned the<br />
"narrow laboratory" of the physicist <strong>and</strong> threw himself at cosmological heights.<br />
He did not give examples of applications of the conservation principle to the<br />
physical <strong>and</strong> chemical laws as he had done in the Erhaltung. Instead he gave a<br />
large-scale, unified picture of the world in energy terms: from the Kant-Laplace<br />
cosmogonic hypothesis to thermal death, to an analysis of the planetary system,<br />
of the fall of meteors, of winds, tides <strong>and</strong> all sorts of technical machines <strong>and</strong><br />
devices. At variance <strong>with</strong> what had happened in 1847, these lectures were<br />
received enthusiastically both by the scientific milieu <strong>and</strong> by the general public.<br />
The first lecture was soon translated by Tyndall <strong>and</strong> published in the<br />
Philosophical Magazine <strong>and</strong> this was useful for spreading <strong>Helmholtz's</strong> fame<br />
across the Channel. Did this new-old approach have to be chosen, after Clausius'<br />
criticisms, to please both the scientists <strong>and</strong> the public ? Would it not have been<br />
easier to popularize energy referring to its only two asserted forms <strong>and</strong> to only<br />
one model for the forces?<br />
But a second shift takes place at the beginning of the seventies:<br />
empirical rather than theoretical aspects are stressed in the enunciation of the<br />
principle. Was this second shift connected <strong>with</strong> a more empiricist methodological<br />
view adopted by Helmholtz in the late sixties? What was <strong>Helmholtz's</strong> relation to<br />
the other scientists who had been <strong>and</strong> still were working along similar, but not at<br />
all identical lines? Some remarks on these questions are related to the discussions<br />
of Mayer's priority which Helmholtz kept introducing into his popular lectures.<br />
We are thus led to the priority debate, where some of these problems receive an<br />
answer.<br />
_______________________<br />
On February the 7th 1854 in Königsberg, Helmholtz gave a popular<br />
lecture on the interaction of natural forces 297 that was soon to become extremely<br />
famous. On the first of June of the same year, in fact, Helmholtz told his father<br />
that a second edition of the lecture had already been requested 298. The lecture<br />
297 Helmholtz, Hermann: Ueber die Wechselwirkung der Naturkräfte un die darauf<br />
bezüglichen neuesten Ermittelungen der Physik. Königsberg: Gräfe & Unzer,1854.<br />
298 Koenigsberger H v H p.124
was soon translated into English by Tyndall <strong>and</strong> appeared in the Philosophical<br />
Magazine 299, was reprinted <strong>with</strong> slight modifications 300 in the first (1871) <strong>and</strong><br />
second (1876) edition of the second volume of the Populären Wissenschaftlichen<br />
Vorträge, appeared, <strong>with</strong> a note on "Robert Mayer's Priorität" 301, in the first<br />
(1884), second (1896) <strong>and</strong> third edition (1903) of the Vorträge und Reden 302.<br />
<strong>Helmholtz's</strong> renewed interest in the conservation of energy was<br />
<strong>with</strong>out doubt due to the controversy <strong>with</strong> Clausius just discussed 303. But the<br />
result is surprising : to the dem<strong>and</strong> for some more popular accounts "of the great<br />
principle that was to underlie the science of the future" 304 Helmholtz answers <strong>with</strong><br />
a magnificent exposition, ranging to cosmological dimensions, of the law of the<br />
correlations of forces <strong>and</strong> not of his own mechanistic view of conservation of<br />
energy based on central forces <strong>and</strong> on the dichotomy between kinetic <strong>and</strong><br />
potential energies! Despite the skillfulness of his counter-attack on Clausius it<br />
seems that Clausius' objections had a great impact on our author. In fact two<br />
other popular lectures on the same argument (in 1861 <strong>and</strong> 1862-64) 305 share the<br />
same approach.<br />
The argument runs as follows, <strong>and</strong> it is worthwhile quoting it at<br />
length:<br />
"The new philosopher's stone of the seventeenth <strong>and</strong> eighteenth<br />
centuries is perpetual motion. Perpetual motion was to produce work out of<br />
nothing, but a point was reached where it could be proved that at least by the use<br />
of pure mechanical forces no perpetual motion could be generated. The idea of<br />
work became identical <strong>with</strong> that of the expenditure of force. How, then, can we<br />
299 "On the Interaction of Natural Forces <strong>and</strong> recent Physical Discoveries bearing on<br />
the same" Phil Mag s.4 75 Suppl.Vol.11 : pp.489-518; quotations here from Tyndall's<br />
translation.<br />
300 See the author's preface in 1871<br />
301 To be discussed below.<br />
302 The English translation was also included in the first edition (1873), <strong>and</strong> in the first<br />
volume of the second (1881) <strong>and</strong> third (New Edition 1893) editions of the Popular Lectures<br />
on Scientific Subjects translated by Atkinson. A French translation appeared in 1869, together<br />
<strong>with</strong> the translation of the Erhaltung (Paris: V.Masson et Fils); more recently an Italian<br />
translation in: Helmholtz, Hermann. Opere . V.Cappelletti ed. Torino: UTET, 1967.<br />
303 Koenigsberger H v H p.120.<br />
304 Ibidem.<br />
305 See below.
measure this expenditure <strong>and</strong> compare it in the case of different machines? In the<br />
case of a watermill <strong>with</strong> an iron hammer, the work must be measured by the<br />
product of the weight into the space through which it ascends. The work<br />
performed by the hammer is determined by its velocity. The motion of a mass<br />
regarded as taking the place of working force is called the living force (vis viva)<br />
of the mass. Living force can generate the same amount of work as that expended<br />
in its production. It is therefore equivalent to this quantity of work. Mathematical<br />
theory has corroborated this for all purely mechanical, that is to say, for moving<br />
forces. After this law had been established by the great mathematicians of the last<br />
century, a perpetual motion, which should make use of pure mechanical forces,<br />
such as gravity, elasticity, pressure of liquid <strong>and</strong> gases, could only be sought after<br />
by bewildered <strong>and</strong> ill-instructed people." 306<br />
But here comes the new problem caused by the conversion processes<br />
in the 19th century:<br />
"But there are still other natural forces which are not reckoned among<br />
the purely moving forces, heat, electricity, magnetism, light, chemical forces, all<br />
of which st<strong>and</strong> in manifold relation to mechanical processes. Here the question of<br />
a perpetual motion remained open." 307<br />
At this stage <strong>Helmholtz's</strong> argument shows the complete ab<strong>and</strong>onment<br />
of one of the two conceptual roots <strong>and</strong> main assumptions of his Erhaltung,<br />
namely the hypothesis of central forces depending only on distances. In fact the<br />
conservation of force is here seen as the correlation of forces through constant<br />
coefficients, based on the acceptance of the impossibility of perpetual motion :<br />
" ..it was asked, if a perpetual motion be impossible, what are the<br />
relations which must subsist between natural forces? Everything was gained by<br />
this inversion of the question. It was found that all known relations of forces<br />
harmonize <strong>with</strong> the consequences of that assumption, <strong>and</strong> a series of unknown<br />
relations were discovered at the same time, the correctness of which remained to<br />
be proved." 308<br />
Contributors to this line of thought were Carnot in 1824 (despite the<br />
incorrect view of the nature of heat), Mayer in 1842 309, Colding in 1843 310, Joule.<br />
306 Helmholtz "Interaction" pp.489-95<br />
307 Helmholtz "Interaction" Pp.495-6.<br />
308 Helmholtz "Interaction" p.498<br />
309 This is the first published appreciation of Mayer's priority. Helmholtz will repeat it<br />
in 1855 (in the "Bericht" see n.2); in1861 ("On the Application of the Law of the <strong>Conservation</strong>
It seems as though the importance of the different approaches disappeared for<br />
Helmholtz:<br />
"...several heads .... generated exactly the same series of reflections" 311.<br />
Helmholtz asserts he played only the following role:<br />
"I myself, <strong>with</strong>out being acquainted <strong>with</strong> either Mayer or Colding, <strong>and</strong><br />
having first made acquaintance <strong>with</strong> Joule's experiments at the end of my<br />
investigation 312, followed the same path. I endeavoured to ascertain all the<br />
relations between the different natural processes, which followed from our<br />
regarding them from the above point of view." 313<br />
From this analysis <strong>Helmholtz's</strong> specific interpretation of energy<br />
conservation completely disappears. This cannot be attributed to a simplification<br />
due to the attempt at popularizing a difficult subject. The reduction of all the<br />
interactions of natural "forces" to only two kinds of energy <strong>and</strong> to one only, well<br />
known, model of force would have simplified the task. This approach is, in my<br />
opinion, motivated by the desire to offer to the public a picture of the new theory<br />
which was not controversial for the specialists; it is the first implicit<br />
acknowledgement of a weakness in <strong>Helmholtz's</strong> position of 1847 <strong>and</strong> the<br />
beginning of a retreat to safer ground.<br />
The subsequent steps of the argument recall: the recognition that work,<br />
apart from not being created, cannot be destroyed; the mechanical theory of heat;<br />
<strong>and</strong> the success in determining the work-heat equivalent. Again <strong>Helmholtz's</strong> own<br />
results are expressed in simplified terms :<br />
of Force to Organic Nature." In Proc Roy Inst 3(1861): 347-57; rep. inW A 3, pp.565-80, at<br />
p.573); in 1862-4 (see: "On the <strong>Conservation</strong> of Force." In Pop Lect 1873, p.320); in a letter<br />
to Tait, published in Tait's Sketch of Thermodynamics in1868; in 1882 in an appendix to the<br />
reprint of the Erhaltung : WA 1 Pp.71-4. But in 1883, after the controversy <strong>with</strong> Dühring, its<br />
judgement on Mayer's contribution was much less appreciative. See below.<br />
310 On Colding see: Dahl, Per. "Ludwig A. Colding <strong>and</strong> the <strong>Conservation</strong> of <strong>Energy</strong>."<br />
In Centaurus 8 (1963): 174-88.<br />
311 Helmholtz "Interaction" p.499<br />
312 I am inclined to think that this is true in view of the lack of reference to Joule in<br />
the 1847 "Bericht" <strong>and</strong> of the little room dedicated to Joule in the fourth chapter of the<br />
Erhaltung compared <strong>with</strong> Clapeyron <strong>and</strong> Holtzmann, surely less relevant to the problem of<br />
the heat-work equivalent.<br />
313 Helmholtz "Interaction" p.499
" ...the quantity of force in nature is just as eternal <strong>and</strong> inalterable as<br />
the quantity of matter. Expressed in this form, I have named the general law 'The<br />
Principle of <strong>Conservation</strong> of Force'." 314.<br />
Clausius is clearly <strong>and</strong> explicitly praised for the first expression of the<br />
second principle, a principle that does not contradict the law of conservation of<br />
force :<br />
"Only when heat passes from a warmer to a colder body, <strong>and</strong> even then<br />
only partially, can it be converted into mechanical work" 315.<br />
After giving credit to Carnot <strong>and</strong> W.Thomson, to the latter also for the<br />
problem of thermal death, Helmholtz ab<strong>and</strong>ons mathematical-mechanical<br />
developments <strong>and</strong> instead of giving "a glance at the narrow laboratory of the<br />
physicist" gives " a glance at the wide heaven above us, the clouds, the rivers, the<br />
woods <strong>and</strong> the living beings around us" 316.<br />
An extraordinary series of applications of the great law now takes<br />
place, ranging from a reassessment of the Kant-Laplace hypothesis (<strong>with</strong> a<br />
calculation of the heat produced by the assumed condensation of the bodies of<br />
our system from scattered nebulous matter) to the evaluation of the heat produced<br />
by the speed of the meteors, <strong>and</strong> to the comparison of the heat coming to the<br />
surface of the earth from <strong>with</strong>in, <strong>with</strong> that reaching the earth from the sun. The<br />
conservation of force is then applied to organic bodies : we can calculate from<br />
the mass of the consumed nutritive material how much heat, or its equivalent<br />
work, is generated in an animal body; the influence of the sun explains why the<br />
combination of the animal <strong>and</strong> vegetable organic realms does not produce<br />
perpetual motion. There is thus a specific sense in which we can all consider<br />
ourselves to be "as the great monarch of China, sons of the sun"! 317 The ebb <strong>and</strong><br />
flow of tides are explained too, through the combined action of the sun <strong>and</strong> the<br />
moon. The motions of the tides, as already done by Mayer 318, are connected to<br />
the law of conservation of force: they "produce friction, all friction destroys vis<br />
viva, <strong>and</strong> the loss in this case can only affect the vis viva of the planetary<br />
system." Finally the thermal death of the sun is discussed.<br />
314 Helmholtz "Interaction" p.501<br />
315 Helmholtz "Interaction" p.502<br />
316 Helmholtz "Interaction" p.503<br />
317 Helmholtz "Interaction" p.511<br />
318 Helmholtz "Interaction" p.512
From the refusal of perpetual motion we have been "conducted to a<br />
universal law of nature, which radiates light into the distant nights of the<br />
beginning <strong>and</strong> of the end of the history of the universe." 319.<br />
Without doubt Helmholtz proved to be a populariser of the first rank<br />
<strong>and</strong> a master not only of physics, physiology <strong>and</strong> cosmology but also of the<br />
German language 320. In the fight against the antiscientific philosophical approach<br />
still present in German universities this talk was to play not a minor role 321. The<br />
capacity of giving a scientific, unified picture of the world in relatively simple<br />
terms <strong>and</strong> <strong>with</strong> the application of only one law was doubtless of great appeal to<br />
the German mind 322.<br />
Helm correctly underlines the stress given in this talk to the ideas of<br />
Grove <strong>and</strong> Mayer on the conversion <strong>and</strong> correlation of forces. This was to be<br />
done by Helmholtz for decades, <strong>with</strong> his ever-growing authority, "also <strong>with</strong>out<br />
dealing <strong>with</strong> his own ideas of 1847" 323.<br />
<strong>Helmholtz's</strong> effort, along the lines just mentioned, was in fact carried<br />
on in other "popular" talks in the following years. Of interest for us is the one<br />
given on the 12th of April 1861 at the Royal Institution in London 324: "On the<br />
Application of the law of the <strong>Conservation</strong> of Force to Organic Nature" 325<br />
For the second time 326 we find here acknowledged, but not <strong>with</strong>out<br />
reservations, the use of the term energy:<br />
"It might be better perhaps to call it, <strong>with</strong> Mr.Rankine, 'the<br />
conservation of energy', because it does not relate to that which we call<br />
commonly intensity of force; it does not mean that the intensity of the natural<br />
319 Helmholtz "Interaction" p.516.<br />
320 Helm, who is not always sympathetic <strong>with</strong> Helmholtz, as already seen, speaks of<br />
"papers <strong>and</strong> talks stylistically <strong>and</strong> pedagogically perfect, even classics". Helm Energetik part<br />
1cap 9 par 7.<br />
321 Koenigsberger H v H p.124.<br />
322 To the British too, judging from the rapidity of the English translation of Tyndall<br />
<strong>and</strong> from the success Helmholtz always had in Britain.<br />
323 Helm Energetik p.<br />
324 This had to be given <strong>with</strong>out any preparation for the insistence of Faraday,<br />
according to Koenigsberger H v H p.199.<br />
325 See n. 3O9.<br />
326 The first being in 1856 : Helmholtz, Hermann. "Bericht über “die Theorie der<br />
Wärme“ betreffende Arbeiten aus dem Jahre 1853". In Fort d Ph 9 (1856): 404-32; p.
forces is constant: but it relates more to the whole amount of power which can be<br />
gained by any natural process, <strong>and</strong> by which a certain amount of work can be<br />
done." 327<br />
The introduction of the term energy does not imply the use of the<br />
dichotomy kinetic/potential energy. As in the previous talk, Helmholtz does not<br />
use his central forces approach <strong>and</strong> its implications, but the conversion of forces<br />
through constant coefficients, <strong>with</strong> the work as unity of measurement of the<br />
different effects:<br />
"But while, by every alteration in nature, that force which has been the<br />
cause of this alteration is exhausted, there is always another force which gains as<br />
much power of producing new alterations in nature as the first had lost" 328.<br />
Examples are given of different forms of motive powers: a raised<br />
weight, velocity, elasticity of a bent spring, elasticity of air <strong>and</strong> of compressed<br />
gases, heat, chemical forces. After a short summary of the cosmological problems<br />
Helmholtz goes back to the priority issue <strong>and</strong> acknowledges the works of Grove,<br />
Joule, Mayer, to which "the first exposition of the general principle" 329 is due.<br />
A remark on the "close connection between both the fundamental<br />
questions of engineering <strong>and</strong> the fundamental questions of physiology <strong>with</strong> the<br />
conservation of force" introduces the main theme of the talk, the application of<br />
the law to organic nature. We can change a certain amount of food into carbonic<br />
acid, water, <strong>and</strong> nitrogen, either by burning the whole in an open fire, or by<br />
giving it to living animals as food <strong>and</strong> burning afterwards only the urea, <strong>and</strong> in<br />
both cases we get the same result. Thus the total equivalent in mechanical work<br />
spent by an animal, that is, the sum of the animal heat <strong>and</strong> the actual mechanical<br />
work, must be equivalent to the heat produced by the burning of the food in the<br />
open fire. For animals at rest Dulong <strong>and</strong> Despretz found that the two quantities<br />
are nearly identical, but some experimental difficulties still exist 330. When work is<br />
performed by animals, for instance ascending a hill, the consumption is five times<br />
greater than when they rest. But only one fifth of this consumption is really spent<br />
in mechanical work, the other four fifths result in the production of heat. An<br />
interesting analogy <strong>with</strong> a thermodynamic engine is drawn: only one eighth of the<br />
equivalent of the chemical force is really converted into mechanical work, the<br />
327 Helmholtz "Organic Nature" p.563.<br />
328 Helmholtz "Organic Nature" p.566.<br />
329 Helmholtz "Organic Nature" p.572.<br />
330 A problem lasting from the 1846 <strong>and</strong> the 1847 papers.
other seven eighths are lost in the form of heat: "the human body is a better<br />
machine than the steam engine, only its fuel is more expensive than the fuel of<br />
steam engines" 331. But despite the fact that we cannot yet prove that the work<br />
produced by living bodies is a total equivalent of the chemical forces which have<br />
been set into action, still "I think we may consider it as extremely probable that<br />
the law of the conservation of force holds good for living bodies" 332. Finally the<br />
"vital principle" is disregarded, in that, if there is complete conservation of force,<br />
the physical forces in the living body cannot be removed <strong>and</strong> again set in action<br />
at any moment by the influence of the vital principle. In fact conservation of force<br />
can exist only in those systems in which the forces in action (like all the forces of<br />
inorganic nature) have always the same intensity <strong>and</strong> direction if the<br />
circumstances under which they act are the same.<br />
Indeed Helmholtz can claim that in a few years the principle has<br />
completely modified the view of life phenomena, <strong>and</strong> that a general unified<br />
approach to physical, cosmological <strong>and</strong> physiological phenomena has been<br />
achieved.<br />
In November 1862 Helmholtz became Pro-Rector of Heidelberg<br />
University <strong>and</strong> on this occasion delivered a talk on "The Relation of Natural<br />
Sciences to Science in General" 333. After analysing the differences of the Kantian<br />
epistemological approach <strong>and</strong> that of Schelling <strong>and</strong> Hegel, Helmholtz recalls <strong>and</strong><br />
exp<strong>and</strong>s some of the ideas already expressed in the Introduction to the 1847<br />
Erhaltung. Natural science cannot be satisfied <strong>with</strong> a collection of facts: it<br />
requires the law that rules them <strong>and</strong> the corresponding causes. There is a<br />
difference between artistic <strong>and</strong> logical induction : free will implies the<br />
impossibility of reducing our psychological expressions to a rigid law. One of the<br />
deep differences between natural sciences <strong>and</strong> human sciences is exactly this :<br />
the former can attain quite general rules <strong>and</strong> laws , the latter judge on the basis<br />
of a psychological sensibility. Human sciences cannot unify observations <strong>and</strong><br />
331 Helmholtz "Organic Nature" p.<br />
332 Ibidem<br />
333 Helmholtz, Hermann. "Ueber das Verhältniss der Naturwissenschaften zur<br />
Gesammtheit der Wissenschaften". Rectoratsrede. Heidelberger Universitätsprogramm 1862.<br />
Rep. in Populäre Wissenschaftliche Vorträge 1st vol. Braunschweig: Vieweg, 1865 (<strong>and</strong><br />
following editions); translated in English as "On the relation of the physical sciences to science<br />
in general" in the Annual Report of the Smithsonian Institution for the year 1871, p.217-234;<br />
reproduced in PL 1873 (<strong>and</strong> following editions); also in R.Kahl SW.
experiences in general laws of unlimited validity <strong>and</strong> extraordinarily great<br />
extension. There is no doubt that looking at nature means to look at a series of<br />
rigorous causal connections, <strong>with</strong>out exceptions. The effort to find a causal<br />
connection everywhere, or to assume it, is one of the features <strong>and</strong> lessons of<br />
natural sciences.<br />
These remarks on causal laws are a leitmotiv of <strong>Helmholtz's</strong> analysis of<br />
energy conservation. He will in fact discuss the topic in 1862, 1864 <strong>and</strong> later<br />
again.<br />
Again the unifying power of reason is at the root of the next series of<br />
lectures, whose introductory talk still bears a famous title: "Ueber die Erhaltung<br />
der Kraft" 334.<br />
Mentioned in a letter to W.Thomson on the 14th of December 1862 335,<br />
they were given in Karlsruhe in the winter of 1862-63 <strong>and</strong> again, as mentioned in<br />
a letter to Ludwig of the 27th of February 1864 336, "in London at Easter in<br />
English" to an audience of "three hundred, <strong>and</strong> among them a number of scientific<br />
men" 337.<br />
<strong>Helmholtz's</strong> point of departure in the talk is completely philosophical,<br />
<strong>with</strong> a strong inclination towards neo-Kantianism. Helmholtz first refers to the<br />
element of distinction between natural <strong>and</strong> mental sciences as outlined in the<br />
previous work 338: the special conformity <strong>with</strong> laws of natural phenomena. It has<br />
334 Helmholtz, Hermann. "Ueber die Erhaltung der Kraft." In Populäre<br />
Wissenschaftliche Vorträge 2nd vol. Braunschweig: Vieweg, 1871. A summary of the English<br />
talks: "Lectures on the <strong>Conservation</strong> of <strong>Energy</strong>. Delivered at the Royal Institution." was<br />
published in the Medical Times <strong>and</strong> Gazette vol 1, 1864; a German version appeared <strong>with</strong> a<br />
preface in the 1871 Populäre Wissenschaftliche Vorträge (<strong>and</strong> following editions), of which<br />
an English translation by E.Atkinson: "On the <strong>Conservation</strong> of Force"was included in 1873<br />
Popular Lectures on Scientific Subjects (<strong>and</strong> following editions). Quotations here are from<br />
this translation. In the preface to the 1884 Vorträge und Reden we are told that of this series<br />
the only other existing revised talk is: "Ueber die Entstehung des Planetensystems" in<br />
Populäre Wissenschaftliche Vorträge 1876, Vol 3 (<strong>and</strong> following editions),translated in<br />
Popular Lectures on Scientific Subjects 1881 (<strong>and</strong> following edition).<br />
p.225.<br />
335 Koenigsberger H v H p.214.<br />
336 Koenigsberger H v H p.220.<br />
337 Helmholtz to Du Bois-Reymond, May 15 1864, quoted in Koenigsberger H v H<br />
338 Helmholtz, Hermann. "On the relation" see n.333.
in fact been possible to discover laws that allow prediction of the origin <strong>and</strong><br />
progress of many extended series of phenomena <strong>with</strong> the greatest accuracy. It is<br />
in this conformity <strong>with</strong> law that the intellectual fascination which links the<br />
physicist to his subject is based. As in mental science so in natural science, a<br />
single fact can provoke curiosity or astonishment, but intellectual satisfaction can<br />
only be given by the conformity <strong>with</strong> law, by the connection of the whole.<br />
The Kantian trend is more <strong>and</strong> more evident : which is the innate<br />
faculty of thought that allows us to discover laws <strong>and</strong> to apply them? "Reason",<br />
of course. And the best arena for the forces of "pure reason " is the inquiry into<br />
nature.<br />
In a very interesting passage Helmholtz makes an assertion of the<br />
greatest importance as to the motivations of a scientist: the reward is not only a<br />
successful activity or the acquisition of power over a sometimes hostile world,<br />
but also the artistic satisfaction of surveying Nature as a regularly ordered<br />
whole, an "image of the logical thought of our own mind" 339.<br />
This regulative power of reason, this highest scientific activity, this<br />
ideal of Kantian <strong>and</strong> neo-Kantian philosophy has now produced a concrete<br />
example that is beneath our eyes : an all-embracing regulative principle, the Law<br />
of the <strong>Conservation</strong> of Force. This law is specially suited to give us an idea of<br />
the specific character of natural sciences. It is of utmost interest then that<br />
<strong>Helmholtz's</strong> enunciation of the law is greatly different from the original one:<br />
"(it) asserts that the quantity of force which can be brought into action<br />
in the whole of Nature is unchangeable <strong>and</strong> can neither be increased nor<br />
diminished" 340.<br />
To explain the meaning of quantity of force, Helmholtz refers to its<br />
technical application, the amount of work, <strong>and</strong> analyses the subject in a way that<br />
recalls the previous popular lectures, but <strong>with</strong> greater technical (not<br />
mathematical) details <strong>and</strong> a generous use of good drawings <strong>and</strong> schemes of<br />
contemporary machines.<br />
<strong>Helmholtz's</strong> appreciation of the work of Mayer <strong>and</strong> Joule in this lecture<br />
is relevant, also for a comparison <strong>with</strong> a later assessment 341. Robert Mayer is here<br />
credited <strong>with</strong> the first statement of the possibility of a universal application of the<br />
law of the conservation of force, <strong>and</strong> Joule <strong>with</strong> having made:<br />
339 Thus there is something more than"the intellectual mastery of nature"!<br />
340 Helmholtz PL 1873 p.320.<br />
341 "Robert Mayer's Priorität" of 1884, see below.
"a series of important <strong>and</strong> difficult experiments on the relation of heat<br />
to mechanical force, which supplied the chief points in which the comparison of<br />
the new theory <strong>with</strong> experience was still wanting" 342.<br />
Joule's experiments are actually given great attention, in contrast <strong>with</strong><br />
the 1847 Erhaltung. This time, of course, the conversion factors are correct. For<br />
the determination of the equivalent through the production of heat from work<br />
utilising friction, three series of experiments are recalled: water in a brass vessel<br />
(equivalent=424.9 gramme-meters necessary to raise of one degree °C one<br />
gramme of water), mercury in an iron vessel (425 <strong>and</strong> 426.3), conical rings<br />
rubbing against each other surrounded by mercury (426.7 <strong>and</strong> 425.6). The reverse<br />
determination, that is, the conversion of heat into work through the expansion of<br />
perfect gases, is also discussed; results (also due to Regnault's measurements)<br />
give : <strong>with</strong> air 426.0; <strong>with</strong> oxygen 425.7; <strong>with</strong> nitrogen 431.3; <strong>with</strong> hydrogen<br />
425.3.<br />
The agreement between the two set of experiments, realised <strong>with</strong> such<br />
different methods is really remarkable. There can be few doubts that "heat is a<br />
new form in which a quantity of work can appear" 343.<br />
A comment on the presentation of the principle is necessary here: the<br />
expression of the principle above 344 shows once again that the stress is on the<br />
correlation of forces, but Helmholtz goes even further in hiding his own<br />
conceptual model of central forces that had played such a great role in the<br />
original Erhaltung . In the extension of the law to all natural processes, the<br />
question of the nature of heat became specially important. "In the answer lay the<br />
chief difference between the older <strong>and</strong> newer views in these respects". Moreover<br />
in virtue of the relevance of this question:<br />
"Many physicists designate that view of Nature corresponding to the<br />
law of conservation of force <strong>with</strong> the name of the Mechanical Theory of Heat" 345.<br />
Thus Helmholtz stresses an element, the conceptual model of heat,<br />
which despite being universally acceptable <strong>and</strong> also accepted, was by no means<br />
his own specific contribution to the problem. After an acknowledgement of the<br />
results of the kinetic theory of gases due to Krönig, Clausius <strong>and</strong> Maxwell 346 <strong>and</strong><br />
342 Helmholtz PL 1873 p.320.<br />
343 Helmholtz PL 1873 p.349.<br />
344 See note 340.<br />
345 Helmholtz PL 1873 p.342.<br />
346 Helmholtz PL 1873 p.350.
a discussion of the production of work through chemical <strong>and</strong> electrical forces, the<br />
paper ends <strong>with</strong> the relations between the principle of conservation <strong>and</strong> perpetual<br />
motion. The ab<strong>and</strong>onment of one of the roots of the 1847 Erhaltung is further in<br />
evidence:<br />
"the possibility of a perpetual motion was first finally negated by the<br />
law of the conservation of force, <strong>and</strong> this law might also be expressed in the<br />
practical form that no perpetual motion is possible, that force cannot be produced<br />
from nothing" 347.<br />
We can conclude that a main intellectual shift happened after the<br />
controversy <strong>with</strong> Clausius regarding the interpretation of energy conservation:<br />
Helmholtz decided to hide in the popular lectures of the 1854-64 period the<br />
central force requirement <strong>and</strong> the dichotomy potential/kinetic energy, to hide also<br />
the theoretical applications of the principle specific to physics (the narrow<br />
laboratory) <strong>and</strong> instead to insist on the cosmological implications <strong>and</strong><br />
technological applications, that is, on the explanation of the way in which<br />
machines work.<br />
But the seventies introduce a second even more relevant<br />
methodological <strong>and</strong> conceptual shift, already outlined by DuBois-Reymond in his<br />
Commemorative Lecture on Helmholtz 348. He remarked on a modification of<br />
methodological approach between the first <strong>and</strong> second part of <strong>Helmholtz's</strong> life:<br />
"just as the principle of the conservation of energy has been a safe clue to<br />
<strong>Helmholtz's</strong> train of thought in the preceding period, so in the later part we have a<br />
similar guide. The fundamental principle of these researches is the empiricist<br />
attitude, which Helmholtz favours in preference to the nativistic, which he<br />
rejected. This is the same contrast that obtained in the sixteenth century between<br />
Leibniz's pre-established harmony <strong>and</strong> Locke's sensualism, <strong>and</strong> to which Kant<br />
gave a decided turn in favour of the former doctrine".<br />
This change of methodology is probably linked <strong>with</strong> the deep new<br />
scientific achievements reached by Helmholtz through tireless activity in the late<br />
sixties. Relevant here are the third part of the H<strong>and</strong>buch der Physiologischen<br />
Optik 349of 1867, the papers "On the Facts that underlie Geometry" 350 <strong>and</strong> " On<br />
1867.<br />
347 Helmholtz PL 1873 p.<br />
348 Koenigsberger H v H pp.237-8<br />
349 Helmholtz, Hermann. H<strong>and</strong>buch der Physiologischen Optik 3 Leipzig: Voss,
the Discontinuous Movements of Fluids" 351of 1868, <strong>and</strong> the long paper<br />
comparing the different electrical theories of 1870 352. The new scientific results<br />
implied among others a reassessment of some aspects of <strong>Helmholtz's</strong> Kantianism.<br />
I will confine myself here to discussing the implications of this new trend for<br />
<strong>Helmholtz's</strong> evaluation of energy conservation.<br />
In September 1869, in Innsbruck, at the Opening of the Natural<br />
Science Congress, Helmholtz gave the address "The Aim <strong>and</strong> Progress of the<br />
Natural Sciences" 353 which once again deals <strong>with</strong> energy conservation. But it is in<br />
1871, in the Preface to the second volume of the Populäre Wissenschaftliche<br />
Vorträge, that it is possible to find the first influences of a new approach to the<br />
energy debates. Helmholtz in fact, recently appointed to the Berlin chair of<br />
physics, introduces the first printed version of the second Erhaltung 354 <strong>with</strong> a<br />
remark on Leibniz <strong>and</strong> Kant. He shows a first detachment from Kant's positions<br />
in the assertion that Kant had misunderstood conservation issues: " the<br />
conference develops a portion of the one on the Interaction of Natural Forces.<br />
This portion deals <strong>with</strong> the fundamental ideas of the subject, still misunderstood<br />
also by people learned in mathematical mechanics. This is not surprising, because<br />
"even such a mind as that of Kant found difficulty in comprehending them, as is<br />
shown by his controversy <strong>with</strong> Leibniz" 355<br />
The influences of <strong>Helmholtz's</strong> methodological shift on the evaluation of<br />
energy debates can be seen in the different judgements that he gave of Mayer's<br />
accomplishments in the popular talks of 1854-64 <strong>and</strong> in two later short works.<br />
These are: the notes to the 1847 Erhaltung written in 1881 <strong>and</strong> published in<br />
1882 in the first volume of the Wissenschaftliche Abh<strong>and</strong>lungen <strong>and</strong> : "Robert<br />
350 Helmholtz, Hermann. "Ueber die Thatsachen, die der Geometrie zu Grunde<br />
liegen." In Nach der k Ges d Wiss zu Gött 9 (1868): 193-221; repr. in WA 2 pp.618-<br />
351 Helmholtz, Hermann. "Ueber discontinuirliche Flüssigkeitsbewegungen." In<br />
Berliner Monatsberichte (1868): 215-28; rep. in W A 1 pp.146- .<br />
352 Helmholtz, Hermann. "Ueber die Bewegungsgleichungen der Elektricität für<br />
ruhende leitende Körper." In Borchardt's Journ 72 (1870): 57-129; repr. in WA 1 pp.545-628.<br />
353 Helmholtz, Hermann. "Ueber das Ziel und die Fortschritte der Naturwissenschaft."<br />
In Populäre Wissenschaftliche Vorträge 2nd vol. Braunschweig: Vieweg, 1871. Tr. in PL<br />
1881.<br />
354 Originally presented in 1862-3 <strong>and</strong> 1864, see above.<br />
355 A reference to Kant's paper on the vis viva controversy:
Mayer's Priorität", an appendix to his "Interaction of Natural Forces" of 1854,<br />
written in 1883 <strong>and</strong> published in the 1884 edition of the Vorträge und Reden.<br />
As already recalled, in1881 Helmholtz moves away from some aspects<br />
of the Kantian approach : he asserts that the philosophical introduction of the<br />
1847 Erhaltung was influenced by Kant to a higher degree than "I would think<br />
legitimate now" 356 .<br />
Helmholtz also acknowledges Mayer's merits 357, <strong>and</strong> explains that he<br />
did not know of his works in 1847, but that from 1854 on he had always given<br />
him due credit. He also quotes at length a letter he wrote to Tait 358 to defend<br />
Mayer's priority. In 1868 Helmholtz still stresses Mayer's theoretical<br />
contributions:<br />
".. the glory of the discovery remains <strong>with</strong> those who have found the<br />
new idea; the experimental proof is, afterwards, a much more mechanical task".<br />
We cannot ask that "the discoverer of the idea be obliged to do also<br />
the second part of the job". Otherwise we should reject also most of the works of<br />
mathematical physicists, including some of Tait's great friend W.Thomson.<br />
The 1868 letter follows the approach of the popular lectures discussed,<br />
in giving due credit to the theoretical aspects of the conservation principle. But<br />
the situation in 1881 is radically different : a bitter controversy had opposed<br />
Helmholtz to Zoellner 359, an astronomer <strong>with</strong> spiritualist inclinations <strong>and</strong><br />
defendent of Weber's electrodynamic law, <strong>and</strong> to Dühring, a historian of<br />
philosophy who had written in 1873 a critical history of the principles of<br />
mechanics 360. The main point of this second controversy was about scientific<br />
356 First appendix to the Erhaltung in WA 1 p.68. Both Kahl's <strong>and</strong> Lindsay's<br />
translations radically differ on this point : for them Helmholtz "still" considers Kant's strong<br />
influence as correct.<br />
357 Fifth appendix to the Erhaltung in WA 1 p.<br />
358 Tait reproduced it in the Introduction to his Sketch of Thermodynamics.<br />
Edinburgh, 1868. This is part of a very long <strong>and</strong> bitter controversy on the priority of the<br />
"discovery" of the conservation principle, mostly on the pages of the Philosophical Magazine<br />
between 1862 <strong>and</strong> 1865. See below.<br />
359 See n.371 below.<br />
360 Dühring for many years claimed that the "guild of professionals" led by Helmholtz<br />
persecuted Mayer. Dühring was also the target of a polemic book of F. Engels. In the end he<br />
lost his teaching position in Berlin University. See: Lindsay, Robert Bruce. Julius Robert<br />
Mayer, Prophet of <strong>Energy</strong>. Oxford <strong>and</strong> New York: Pergamon Press,1973; pp.12-16.
methodology. Dühring claimed the character "a priori" of the conservation law<br />
<strong>and</strong> was one of those who recognised in Mayer "a hero in the field of pure<br />
thought" 361.<br />
Helmholtz after these controversies shows a second shift from his own<br />
positions of 1847: now not only the conceptual explanation of central forces is<br />
given up, but also the theoretical character of the conservation principle: the<br />
empirical component acquires greater <strong>and</strong> greater importance.<br />
In fact the judgement on Mayer is radically different. Mayer's papers'<br />
weakness is exactly what is praised by metaphysicians:<br />
"the illusory demonstration, metaphysically formulated, of the a priori<br />
necessity of this law".<br />
The law 's success was due to Joule's results. Only then was attention<br />
paid to Mayer's work. The origins of the law, even those of the theoretical<br />
demonstration, are inductive: they came from the empirical acknowledgement of<br />
the impossibility of perpetual motion. The (1775) statement of the Paris Academy<br />
was based on such a probable inductive "conviction" (<strong>and</strong> not on a<br />
demonstration). The conviction was largely shared. Helmholtz himself, he recalls,<br />
since his school years, had heard discussions 362 on the problems of proving<br />
perpetual motion. His target in writing the Erhaltung was not to propose an<br />
original idea but to carry out a critical work, <strong>and</strong> he was thus surprised that only<br />
Jacobi, "the mathematician", received it well.<br />
Later, in 1883, while asserting, against Dühring, that he had been the<br />
first in 1854 to recognize Mayer's priority, Helmholtz again attacks the<br />
metaphysicians, who believe the conservation law to be an a priori knowledge.<br />
This is the real point of the debate, not Mayer's priority: the problem is the old<br />
fight between "speculation <strong>and</strong> empiricism", deduction <strong>and</strong> induction. In this<br />
light, any special concern for Mayer's personal difficulties should be overcome<br />
<strong>and</strong> the history of the law of conservation of force clearly stated. The<br />
impossibility of perpetual motion had already been stated in the last century for<br />
"conservative forces" after the works of Leibniz <strong>and</strong> Daniel Bernoulli. Helmholtz<br />
agrees explicitly <strong>with</strong> Mayer's identification of the two philosophical roots of the<br />
conservation principle: the "ex nihilo nil fit " <strong>and</strong> the "nil fieri ad nihilum". He<br />
asserts that the first was thus already accepted in the previous century, <strong>and</strong> that<br />
the second, equivalent to the assertion that work could not be destroyed, despite<br />
361 Helmholtz WA 1 p.<br />
362 See also Koenigsberger H v H p.8 <strong>and</strong> 25-6.
not being explicitly formulated, was already in the background (of the vis viva<br />
conservation) for conservative forces: in this case work cannot be destroyed. The<br />
problem of the nature of heat helped, through many difficulties, to gain wide<br />
acceptance for the second philosophical root. At the beginning of the forties the<br />
existence of the caloric was widely questioned, not only in scientific debates:<br />
Helmholtz himself recalls an essay on the topic that he had to write when a<br />
student at the Potsdam Gymnasium. Apparently here Helmholtz forgets his own<br />
difficulties in giving up <strong>with</strong> the caloric in his first papers. He again defines his<br />
Erhaltung not as an original research but as a critical survey, aimed at clarifying<br />
<strong>and</strong> reordering known facts <strong>and</strong> deciding between different explanations.<br />
Mayer was not, then, completely original even if his paper of 1842<br />
deserves due credit. But the credit is for the priority in the assertion of an idea,<br />
not for his demonstration. Without an empirical demonstration of the<br />
indestructibility of forces, the " deduction" from obscure metaphysical principles<br />
as "causa aequat effectum" is worthless. Mayer did not give this empirical<br />
evidence in 1842 (he did not explain but only asserted that the work equivalent<br />
was 365 kgm). But even if he had explained his calculation of the work<br />
equivalent he would not have shown anything: "it was necessary to show that<br />
different processes give the same value". The accomplishment of this great<br />
experimental task is the lasting merit of Joule.<br />
Helmholtz becomes more <strong>and</strong> more severe: only after Joule's work do<br />
Mayer's views acquire the character of "non improbable hypotheses" 363 <strong>and</strong><br />
Mayer's destiny teaches young researchers that the best ideas risk to be sterile<br />
<strong>with</strong>out convincing demonstrations.<br />
The priority debate was actually a debate on scientific methodology, as<br />
almost explicitly asserted by Helmholtz. Both in the undervaluation of his own<br />
contribution of 1847 <strong>and</strong> of Mayer's results Helmholtz clearly shows that in the<br />
early eighties the preoccupations <strong>with</strong> an empiricist viewpoint were very strong.<br />
Was this due to the bitter controversy that even led him into a depressive state 364<br />
or are its roots to be found in real scientific problems resulting from his previous<br />
theoretical commitments? <strong>Helmholtz's</strong> original approach to energy conservation<br />
had in fact been seriously challenged in the electrodynamics debates of the<br />
seventies.<br />
363 Helmholtz<br />
364 Koenigsberger H v H p.306.
The first history of the conservation principle: "evolution <strong>and</strong><br />
development, not discovery" (1862-65)<br />
In the pages of the Philosophical Magazine for the years 1862-65<br />
there is a bitter polemic between Tyndall on one side <strong>and</strong> Tait (specially) <strong>and</strong><br />
W.Thomson (mostly in the background) on the other 365. Other participants were<br />
Joule, Colding, Verdet, Bohn, Rankine, Akin. The summary of this controversy<br />
can well be considered one of the first contributions to the historiography of the<br />
principle of conservation of energy. With various qualifications, agreements <strong>and</strong><br />
disagreements, <strong>with</strong> the papers' titles ranging from History of <strong>Conservation</strong> of<br />
<strong>Energy</strong>, to History of Mechanical Theory of Heat, of Dynamical Theory of Heat,<br />
of Thermodynamics, of Energetics, of Force it can be asserted that at the time<br />
there was awareness of the following contributions to the principle, which "was<br />
not discovered but evolved <strong>and</strong> developed":<br />
a) conservation of vis viva: Descartes, Huygens, Leibniz, John<br />
Bernoulli, James Bernoulli, Daniel Bernoulli, D'Alembert, Fresnel, L.Carnot<br />
b) dynamical theory of heat : Bacon, Locke, Rumford, Davy, Young<br />
c) correlation of forces : Rumford, Haldat, Morosi, Seguin, Placidus<br />
Henrich, Mohr, Faraday, Liebig<br />
d) mechanical equivalent : S.Carnot, Clapeyron, Holtzmann, Mayer,<br />
Colding, Joule<br />
e) generalization of the principle : Helmholtz, Clausius, Rankine,<br />
W.Thomson.<br />
What was the assessment of <strong>Helmholtz's</strong> work in this debate? He did<br />
not receive special attention: the focus of the debate was a comparison of the<br />
relative merits of Mayer, defended by Tyndall, <strong>and</strong> Joule defended by Tait <strong>and</strong> by<br />
365 See also: Lloyd, J. "Background to the Joule-Mayer Controversy." In Notes <strong>and</strong><br />
Records of the R.S. 25 (1970): 211-25.
himself. Helmholtz is seen rather as a source of reliable opinions: his (wrong)<br />
1847 judgement that Joule's works of 1843-5 were not completely reliable is<br />
recalled; he is credited <strong>with</strong> being the first who generalized the principle to a<br />
number of applications; it is recognised that in 1854 he was the first to give due<br />
credit to Mayer <strong>and</strong> the first to indicate Mayer, Colding <strong>and</strong> Joule as pioneers . In<br />
this debate no precise evaluation of <strong>Helmholtz's</strong> work is made, nor is he involved<br />
in priority controversies <strong>with</strong> the British, as happened to Mayer <strong>and</strong> Clausius.<br />
This is perhaps due to his good relations <strong>with</strong> both contenders.<br />
___________________________<br />
As already mentioned, Helmholtz met Tyndall in Berlin in 1853 <strong>and</strong>,<br />
thanks to the latter's translation of the Erhaltung , already in 1853, during his first<br />
trip to Britain, he realized he was better known there than at home. <strong>Helmholtz's</strong><br />
fame grew after Tyndall's translation of <strong>Helmholtz's</strong> popular talk of 1854 "On the<br />
Interaction of Natural Forces". An indication is the success, at the time of the<br />
controversy, of the two popular talks on the subject of energy conservation that<br />
Helmholtz gave in London, at the Royal Institution, in 1861 <strong>and</strong> 1864. The talks<br />
had been organised by Faraday <strong>and</strong> Tyndall respectively.<br />
Helmholtz <strong>and</strong> Tyndall kept interacting:in 1870 Helmholtz edited the<br />
German translation of Tyndall's Faraday as a Discoverer (to which he added an<br />
introduction 366), in 1871 Heat as a Mode of Motion , in 1874 the Lectures on<br />
Sound , in 1874 the Fragments of Science (<strong>with</strong> the preface:"On the Attempt to<br />
Popularize Science" 367).<br />
But Helmholtz since his second trip 368 to Britain, in 1855, had become<br />
a good friend also of W.Thomson, whom he met often. For instance in 1864,<br />
366 Helmholtz, Hermann. Vorrede zur deutschen Übersetzung von J.Tyndall Faraday<br />
as a discoverer. Braunschweig: Vieweg, 1870; pp.V-XI.<br />
367 Helmholtz, Hermann. Vorrede und Kritische Beilage zur deutschen Uebersetzung<br />
von J.Tyndall Fragments of Science. Braunschweig: Vieweg, 1874; pp.V-XXV <strong>and</strong> 581-97.<br />
Repr. in V u R 1884. Tr in: Nature 10 (1874): 299-302.<br />
368 Helmholtz was a good traveller; among his trips: in 1840 <strong>and</strong> 1842 in Germany; in<br />
1851 in Germany <strong>and</strong> Italy; in 1853, 1855, 1861, 1864, 1871, 1884 to Britain; in 1866 <strong>and</strong><br />
1881 to Paris; in 1878, 1881 <strong>and</strong> 1883 to Italy; in 1880 to Spain; in 1881 to Austria, in 1893<br />
the dramatic one to Chicago; plus a number of holidays in Pontresina, Switzerl<strong>and</strong>, a meeting<br />
point for other scientists, such as the Italian Blaserna.
while in Britain for the above talks at the Royal Institution, he managed to visit<br />
the Thomsons in Glasgow 369.<br />
Helmholtz also edited the German translation of Thomson <strong>and</strong> Tait's<br />
Textbook of Theoretical Physics . The first part of the first volume appeared <strong>with</strong><br />
an introduction in 1871 370. The second part in 1874 was accompanied by<br />
<strong>Helmholtz's</strong> answer 371 to Zoellner's attacks (partly dealing <strong>with</strong> the debate on<br />
energy conservation). In 1885 <strong>Helmholtz's</strong> review of the first two volumes of<br />
W.Thomson "Mathematical <strong>and</strong> Physical Papers" appeared in Nature 372. It<br />
included an interesting account of the Carnot-Joule-W.Thomson-Clausius ideas<br />
on the dynamical theory of heat <strong>and</strong> on the dissipation of energy.<br />
Less friendly was <strong>Helmholtz's</strong> relationship <strong>with</strong> Tait. In 1868 Tait<br />
published in his Sketch of Thermodynamics a critical letter of <strong>Helmholtz's</strong><br />
underlining Mayer's merits in the formulation of energy conservation 373. In the<br />
same book he credited Helmholtz <strong>with</strong> being "one of the most successful of the<br />
early promoters of the science of energy on legitimate principles" 374, but at the<br />
same time he criticized the central force hypothesis: "...is not in the present state<br />
of experimental science more than a very improbable hypothesis" 375. Helmholtz is<br />
instead credited <strong>with</strong> successful applications of the principle 376 . Tait's judgement<br />
is somehow modified in 1876 in the second edition of the Lectures on some<br />
recent advances in Physical Science, where the two assumptions of Helmholtz ,<br />
now "one of the foremost of living mathematicians <strong>and</strong> natural philosophers" 377,<br />
369 And also to visit Oxford <strong>and</strong> Stokes at Cambridge.<br />
370 Helmholtz, Hermann. Vorrede zum ersten Theil des ersten B<strong>and</strong>es der deutschen<br />
Uebersetzung von: W.Thomson und P.G.Tait Treatise on Natural Philosophy. Braunschweig:<br />
Vieweg, 1871; pp.X-XII.<br />
371 Helmholtz, Hermann: Kritisches. Vorrede zum zweiten Theile des ersten B<strong>and</strong>es<br />
von W.Thomson <strong>and</strong> P.G.Tait Treatise on Natural Philosophy Braunschweig: Vieweg, 1874;<br />
pp.V-XIV. Tr. in Nature<br />
372 Helmholtz, Hermann. "Report on Sir William Thomson's Mathematical <strong>and</strong><br />
Physical Papers". Vol.1 <strong>and</strong> 2. In Nature 32 (1885): 25-8.<br />
achievements.<br />
373 See previous section for <strong>Helmholtz's</strong> changing evaluation of Mayer's<br />
374 Tait Sketch p.<br />
375 Tait Sketch par.96 p.56.<br />
376 Tait Sketch par.104 p. <strong>and</strong> par.127 p..<br />
377 Tait Advances p.67.
are considered equivalent. Still, the separation of energy into two main forms,<br />
potential <strong>and</strong> kinetic, is attributed to Rankine <strong>and</strong> W.Thomson, following a<br />
British nationalist historiographic tradition. In any case Tait's stress is on the<br />
empirical elements of <strong>Helmholtz's</strong> approach: even the assumption of the<br />
impossibility of perpetual motion for the whole of natural forces is judged an<br />
experimental result.<br />
This insistence on the historiographical revaluation of the experimental<br />
elements in the various formulations of the principle is shared in the early eighties<br />
by Helmholtz himself, as seen in the previous section. This is, in my view, owing<br />
to some concrete difficulties faced by Helmholtz in applying his 1847 formulation<br />
of the principle to actual physical research in the late sixties <strong>and</strong> seventies.<br />
A new tool : potential theory (1852-72)<br />
<strong>Helmholtz's</strong> Erhaltung did not <strong>with</strong>st<strong>and</strong> the criticisms pointed out by<br />
Clausius on the definition of potential <strong>and</strong> on the privilege accorded to central<br />
Newtonian forces. Its electrodynamic part was also "for the most part<br />
defective" 378 for its lack of terms referring to energy stored <strong>with</strong> the circuits. The<br />
very derivation of Neumann's law of induction, for which Helmholtz <strong>and</strong><br />
W.Thomson had received so much credit (not the least from Tait in 68-69 <strong>and</strong><br />
Maxwell in 1873), was not correct 379.<br />
Helmholtz realized this before his critics. He acknowledged the<br />
weakness of the electrodynamic aspects of the Erhaltung during the controversy<br />
<strong>with</strong> Clausius <strong>and</strong> defended his own theoretical approach <strong>with</strong> the introduction<br />
of some new energy terms, but from that moment on his research in the field of<br />
energy shifted a great deal. He was in fact more inclined towards the physicomathematical<br />
problems connected <strong>with</strong> Green's theorem <strong>and</strong> <strong>with</strong> the<br />
mathematical theory of potential as heuristic tools, rather than <strong>with</strong> the<br />
theoretical applications in the 1847 style. Helmholtz worked hard to master<br />
potential theory, <strong>and</strong> in the end achieved an extraordinary success: he was able to<br />
378 Whittaker, E. A History of the Theories of Aether <strong>and</strong> Electricity 2nd ed. New<br />
York: Harper, 1960; p.218.<br />
379 See nn.217-221.
apply it to acoustics (1859), hydrodynamics (1859 <strong>and</strong> 1868) <strong>and</strong><br />
electrodynamics (1861, but mainly from 1869-1870 on) in works that became<br />
masterpieces in their fields.<br />
_______________________<br />
In the German-speaking world the mathematical potential (potential<br />
function) <strong>and</strong> the physical potential (potential) spread rapidly <strong>and</strong> were widely<br />
used. Clausius had a prominent role in this, through the different editions of his<br />
treatise on "Potential <strong>and</strong> Potential function", but already F.Neumann in 1845,<br />
Weber <strong>and</strong> Kirchhoff in 1848 <strong>and</strong> Riemann had made various use of these<br />
concepts. C.Neumann too followed this line of thought; in the sixties <strong>and</strong><br />
seventies delayed potentials <strong>and</strong> generalised potentials were also introduced.<br />
Given the now generally applied identification of potential <strong>with</strong> the work done, in<br />
this tradition all that was required to satisfy energy conservation was to show that<br />
the force in question (the problems were mainly connected <strong>with</strong> the attempt to<br />
formulate a general electrodynamic force) admitted a potential, that is that the<br />
work done by the force was a total differential. This in fact is equivalent to<br />
admitting the impossibility of perpetual motion.<br />
In Britain the situation was somewhat different. Despite the early <strong>and</strong><br />
important contributions to potential theory by Green, Hamilton, W.Thomson <strong>and</strong><br />
Stokes, priority was in general given to the "classic" concept of energy, implying<br />
the clear distinction between potential <strong>and</strong> kinetic parts. Of these two the kinetic<br />
was to be seen as basic : J.J.Thomson among others attempted to show this<br />
supposed priority, while Maxwell in 1873 had defined the idea of a delayed<br />
potential as "inconceivable".<br />
An energy battlefield for the electrodynamic debate (1870-75)<br />
In Electrodynamics Helmholtz opposed the views of Weber, who since<br />
1846 had given a force law depending not only on distances but also on velocities<br />
<strong>and</strong> accelerations, <strong>and</strong> of Clausius who in 1875-6 published a contribution along<br />
similar lines. Various physicists contributed to various aspects of the debate, but<br />
here we will only refer to the role of the energy principle. Not a minor one<br />
indeed, but rather the main battleground. Given the mathematical equivalence of<br />
the competing theories <strong>and</strong> the difficulties in experimenting <strong>with</strong> open currents,<br />
the acceptance of one law or the other began to be evaluated in relation to the
expression provided for the conservation principle 380. This is already a<br />
methodological shift of great interest in scientific debates. But the interest is<br />
augmented by the fact that it implied the formulation of more sophisticated<br />
versions of the principle itself. The admissibility of Weber's law had been<br />
questioned on the grounds that his force law did not match <strong>Helmholtz's</strong> criterion<br />
of central forces. Accepting the supposed equivalence of 1847 between central<br />
Newtonian forces <strong>and</strong> the impossibility of perpetual motion, the conclusion was<br />
drawn that Weber's law clashed <strong>with</strong> this basic assumption. Remarks of this kind<br />
were still put forward in the sixties in Britain by Maxwell (1864), W.Thomson<br />
<strong>and</strong> Tait (1867), Tait (1868). The above mentioned fact that Weber's force<br />
admitted a potential was unnoticed, or, given that this potential included both<br />
kinetic <strong>and</strong> positional terms, was still considered to violate the requirement of a<br />
sharp distinction between kinetic <strong>and</strong> potential energy. Finally both Helmholtz<br />
(1870 <strong>and</strong> 1872) <strong>and</strong> Maxwell (1873) agreed that Weber's law was not in<br />
contradiction <strong>with</strong> the impossibility of perpetual motion. But Helmholtz, who<br />
himself had proposed a very general potential law for electrodynamics, kept<br />
criticizing Weber on energy grounds: in Weber's theory energy could become<br />
infinite (1870), kinetic energy could become negative (1872) <strong>and</strong> so forth till<br />
1882. Weber defended himself vigorously in 1871, reformulating the principle of<br />
conservation of energy <strong>and</strong> explicitly stating he did not necessarily have to fulfil<br />
<strong>Helmholtz's</strong> version. He received Carl Neumann's (1871, 1875, 1877) reasonable<br />
support <strong>and</strong> Zöllner's unreasonable one ( 1876). On another side Clausius very<br />
plainly states that the only condition to fulfil energy conservation is that the work<br />
produced by the force be a total differential (1876) <strong>and</strong> not that forces be central<br />
or that kinetic <strong>and</strong> positional terms of energy be sharply split. It is to be noted<br />
that Clausius' target was not so much Helmholtz, but Weber, an indication of the<br />
latter's important role in electrodynamics. <strong>Helmholtz's</strong> electrodynamic potential<br />
law, based on action at a distance <strong>and</strong> on the polarization of an interposed<br />
dielectric, in the seventies began to be seen as an underst<strong>and</strong>able translation of<br />
Maxwell's difficult <strong>and</strong> sometimes contradictory approach. An example of the<br />
difficulties of Maxwell's Treatise (1873) are the many different concepts of<br />
energy presented: as potential energy (electrostatic) <strong>and</strong> kinetic (electromagnetic)<br />
or as the product of an intensity by a quantity factor; electrostatic energy seen as<br />
charge by scalar potential or as the square of the electric intensity;<br />
electromagnetic energy seen as current by vector potential or as the square of the<br />
380 See my <strong>Energy</strong> <strong>and</strong> Electr
magnetic intensity. Maxwell realized the need for a specific interpretation of the<br />
energy concept suited to a contiguous action theory <strong>and</strong> based on the continuity<br />
equation. Helmholtz translated Maxwell's approach in his action at a distance<br />
<strong>with</strong> polarized dielectric approach but was never able, even in 1894, to grasp<br />
Poynting's new concept of localized conservation (1884-5) or Hertz's efforts at<br />
"purifying" Maxwell's theory from action at a distance influences (1890-2). The<br />
interpretation of potential energy in <strong>Helmholtz's</strong> theory was <strong>and</strong> still is a<br />
challenge. J.J.Thomson in 1885 had based his comparison of electrodynamic<br />
theories on energy grounds <strong>and</strong> Planck had been very quick at assessing in 1887<br />
the whole problem of energy conservation, both historically <strong>and</strong> logically.<br />
Helmholtz could not have failed to notice these developments, but at this stage he<br />
was shifting towards a different research programme.<br />
______________________________<br />
In the seventies Helmholtz returned to a problem he had first faced in his 1847<br />
paper: the application of the principle of energy conservation (PCE) to<br />
electrodynamics. From 1847 to 1870 radical changes occurred both in the field of<br />
Classical Electromagnetic Theory (CET) <strong>and</strong> in PCE. Helmholtz’s purpose in<br />
1870 was to reorder the field of CET 381, <strong>with</strong> a clarification of the debate on the<br />
basis of considerations referring to conservation of energy. This attempt lasted<br />
for at least twenty years <strong>and</strong> was in the end successful: Helmholtz in fact played a<br />
major role in comparing the different theories, suggested a set of experiments that<br />
were to become extremely famous <strong>and</strong> gave an initial impulse to Lorentz's<br />
researches.<br />
The complexity of the CET debate at the beginning of the seventies was later<br />
described by Helmholtz himself:<br />
"This plentiful crop of hypotheses had become very unmanageable, <strong>and</strong> in dealing<br />
<strong>with</strong> them it was necessary to go through complicated calculations, resolutions of<br />
forces into their components in various directions, <strong>and</strong> so on. So at that time the<br />
domain of electromagnetics had become a pathless wilderness. Observed facts<br />
<strong>and</strong> deductions from exceedingly doubtful theories were inextricably mixed up<br />
together. With the object of clearing up this confusion I had set myself the task of<br />
381 Helmholtz,Hermann. Vorwort zu: Heinrich Hertz, Prinzipien der Mechanik.<br />
Leipzig: Barth,1894; pp. VII-XXVII. Tr. in Hertz, H. The Principles of Mechanics,<br />
Presented in a New Form. Tr. by D.E.Jones <strong>and</strong> T.Walley. London: 1899. Repr. New York:<br />
Dover,1956. P.4.
surveying the region of electromagnetics, <strong>and</strong> working out the distinctive<br />
consequences of the various theories, in order, wherever that was possible, to<br />
decide between them by suitable experiments." 382.<br />
In fact, in 1870, he started publishing a series of papers that constituted a<br />
comprehensive study of electrodynamics. "With the object of clearing up this<br />
confusion", he gave a theory, whose mathematical version included Weber, F.<br />
Neumann <strong>and</strong> Maxwell’s theory as limiting cases. Helmholtz’s starting point is<br />
relevant: it is the assumption of the existence not only of a force between current<br />
elements but of a force <strong>and</strong> a torque. Both these assumptions contrast <strong>with</strong><br />
Ampére’s theory. Ampére admitted a potential only for closed currents <strong>and</strong> only<br />
a central force between current elements. Helmholtz’s starting point was a<br />
generalisation of F. Neumann’s theory: he gave the most general expression for<br />
the energy of two current elements, consistent <strong>with</strong> the condition that the force<br />
between two closed circuits should be the one given by Ampere:<br />
where A is a constant depending on the unit of current, r the distance between the<br />
elements of circuit Ds <strong>and</strong> Dσ traversed by the currents i <strong>and</strong> j. K too is a<br />
constant, for K=+1 Helmholtz’s potential reduces to F. Neumann’s; for K=-1,<br />
Weber’s potential is obtained; for K=0 plus a dielectric medium, Maxwell’s<br />
theory can be obtained 383. An expression giving both F. Neumann’s <strong>and</strong> Weber’s<br />
electrodynamic energies is:<br />
where h is an arbitrary constant. The first term is Neumann’s value. The second<br />
term is the difference between Weber’s <strong>and</strong> Neumann’s values. The second term<br />
vanishes for closed currents, the only case analysed by F. Neumann 384.<br />
In the fourth paragraph Helmholtz analyses the different theories <strong>with</strong> respect to<br />
energy values 385. The whole energy Φ is the sum of the electrodynamic<br />
382 Ibidem<br />
383 Helmholtz, Hermann. "Ueber die Bewegungsgleichungen der Elektricität für<br />
ruhende leitende Körper." In Borchardt's Journ 72 (1870): 57-129; repr. in WA 1 pp.545-628.<br />
Pp. 549 <strong>and</strong> 567.<br />
384 Ibidem pp.565-6.<br />
385 Ibidem pp.579-85.
<strong>and</strong> the electrostatic<br />
In modern notation 386:<br />
In the absence of electromotive forces external to the system:<br />
is essentially negative. In this case, if K is negative Φ might be negative <strong>and</strong> thus<br />
a perturbation of the system might lead away from equilibrium. That is, the<br />
equilibrium is unstable for K
solve the difficulty of the elastic-solid theory of reflection <strong>and</strong> refraction of light.<br />
This footnote was the starting point of H. Lorentz’s 1875 doctoral thesis. A clear<br />
schematization of the physical model of Helmholtz’s theory was given by Hertz<br />
in 1892 388. Hertz distinguished two limiting cases, both refer to a conception of<br />
action-at-a-distance starting from the charges on the plates that polarize the<br />
dielectric between the plates. In the first case, the energy is supposed to be<br />
concentrated on the plates <strong>and</strong> the actions (at a distance) produced by the<br />
polarised dielectric are comparatively small. Ideally, removing B leaves some<br />
action-at-adistance. In the second case, the energy is supposed to be concentrated<br />
in the dielectric particles <strong>and</strong> so the dielectric action is the prevailing one. This<br />
second case is mathematically equivalent to Maxwell’s. Helmholtz himself was<br />
quite clear about the physical differences between his <strong>and</strong> Maxwell’s theory:<br />
"The two theories are opposed to each other in a certain sense, since according to<br />
the theory of magnetic induction originating <strong>with</strong> Poisson, which can be carried<br />
through in a fully corresponding way for the theory of dielectric polarization of<br />
insulators, the action-at-a-distance is diminished by the polarization, while<br />
according to Maxwell’s theory on the other h<strong>and</strong> the action-at-a-distance is<br />
exactly replaced by the polarization" 389<br />
<strong>and</strong><br />
"It follows from these investigations that the remarkable analogy between the<br />
motion of electricity in a dielectric <strong>and</strong> that of the light aether does not depend on<br />
the particular form of Maxwell’s hypothesis, but results also in a basically similar<br />
fashion if we maintain the older viewpoint about electrical action-at-adistance"<br />
390.<br />
Helmholtz’s comparison between the three theories had already led to<br />
remarkable results: he judged Weber’s theory untenable (for the instability of<br />
the equilibrium given by K < 0) <strong>and</strong> instead showed great interest in Maxwell’s<br />
limit of his own theory. In addition, he considered this last approach suitable to<br />
solve the difficulty of optical theory.<br />
For my purposes, it is relevant to notice <strong>Helmholtz's</strong> shift of criticism towards<br />
Weber’s law. Weber’s law is no longer considered inconsistent <strong>with</strong> the principle<br />
388 Hertz, H. Untersuchungen über die Ausbreitung der elektrischen Kraft. Leipzig,<br />
1892. Tr. by D.E.Jones Electric Waves. London: Mac Millan, 1892. Repr. New York: Dover,<br />
1962. Pp.23-5.<br />
389 Helmholtz "Beweg Elektr" pp.556-7.<br />
390 Helmholtz "Beweg Elektr" p.558.
of conservation of energy: it is recognised that the work done during a complete<br />
cyclical operation is zero, in other words the existence of a potential is<br />
recognized. What is criticized now is the possibility of deducing absurd<br />
consequences from the law. In 1872, Helmholtz modified his PCE <strong>with</strong> the<br />
explicit requirement that T > 0 (T = kinetic energy), in order to theoretically<br />
disregard the supposed inconsistencies derivable from Weber’s law 391.<br />
What was Weber answer? As mentioned in previous sections, in 1846 Weber<br />
published his non-positional force law <strong>and</strong> in 1848 he published a derivation of<br />
this law from the expression of a potential. Both these results contradicted<br />
Helmholtz’s 1847 formulation of PCE. In Weber’s case the forces were not<br />
central <strong>and</strong> there was no clear distinction between kinetic <strong>and</strong> potential energy.<br />
So the problem was: had Weber’s law to be considered in agreement <strong>with</strong> a<br />
general PCE, despite its contrast <strong>with</strong> <strong>Helmholtz's</strong> PCE? Still in 1867 Thomson<br />
<strong>and</strong> Tait gave a negative answer to this question:<br />
"... the conclusion (of Weber’s theory) would st<strong>and</strong> in contradiction <strong>with</strong> the<br />
’<strong>Conservation</strong> of <strong>Energy</strong>’ which we take to be a general law of nature from<br />
innumerable experiments. Such theories are all the more dangerous if they<br />
accidentally explain other phenomena, as Weber’s explain induced currents" 392.<br />
Tait again in 1868:<br />
"But the investigations of these authors (Riemann <strong>and</strong> Lorenz) are entirely based<br />
on Weber’s inadmissible theory of the forces exerted on each other by moving<br />
electric particles, for which the conservation of energy is not true, while<br />
Maxwell’s result is in perfect consistence <strong>with</strong> that great principle." 393.<br />
Tait’s (<strong>and</strong> Thomson’s) objection refers to the contradiction between Weber’s<br />
law <strong>and</strong> Helmholtz’s PCE, but cannot be accepted as an objection that Weber’s<br />
law is incompatible <strong>with</strong> the impossibility of perpetual motion. The existence of a<br />
potential for Weber’s law implies that the work done by the electric forces is a<br />
perfect differential. This means that in a cyclical process an indefinite amount of<br />
work cannot be generated by a particle moving under the action of the force<br />
assumed by Weber. Thus work is not created out of nothing. Finally both<br />
Helmholtz (in 1870 <strong>and</strong> 1872) <strong>and</strong> Maxwell (in 1873) agreed on this important<br />
391 Helmholtz, Hermann. "Ueber die Theorie der Elektrodynamik." Berlin Monats<br />
April 1872: 247-56. Repr. in WA 1 : 636-46. Pp.645-6.<br />
1877. P.318.<br />
392 Quoted in Neumann, Carl. Die Gesetze von Ampére und Weber. Leipzig: Teubner,<br />
393 Tait,P.G. Sketch of Thermodynamics. Edinburgh, 1868. P.76.
point: as far as a potential exists, the corresponding force law does not contradict<br />
the impossibility of perpetual motion. Since this is the first root of the principle of<br />
conservation of vis viva analysed in section 3, it is to be asserted that Weber’s<br />
law originates directly from this root. This approach to PCE was often called the<br />
law of potential 394.<br />
Weber in 1871 proposed his own version of the principle of energy<br />
conservation 395. At the end of the second part of his 1871 paper Weber answers<br />
Helmholtz’s criticisms of 1870 about the incompatibility of Weber’s force law<br />
<strong>with</strong> some consequences of PCE. This was the starting point of the second part of<br />
the lengthy controversy. Weber first notes the new requirement Helmholtz<br />
imposes on PCE. The vis viva cannot become infinite, or otherwise an infinitely<br />
great amount of work could be performed (either in passing from a finite velocity<br />
to an infinite, or from an infinite to a finite). Thus a limiting velocity must exist.<br />
For Weber this limiting velocity is his constant cw=⎟2 vel.light. In Weber’s<br />
view, Helmholtz’s criticism has to be rejected for it assumes an initial velocity of<br />
the particles greater than c. Assuming this limiting value Weber’s potential is<br />
always a positive quantity. In the second place, Weber notes, the finite distance<br />
at which the particles in Helmholtz’s objection would acquire an infinite velocity<br />
is extremely small, outside the domain of enquiry. Thus the objection is<br />
practically meaningless. But Helmholtz again in 1872 <strong>and</strong> 1873 criticised<br />
Weber’s law because of the negative sign in one of the terms of the generalised<br />
potential. In fact this implies that a charge behaves "somewhat as if its mass were<br />
negative, so that in certain circumstances its velocity might increase indefinitely<br />
under the action of a force opposed to the motion" 396. Maxwell in 1873 agreed<br />
<strong>with</strong> Helmholtz’s 1872 criticisms <strong>and</strong> asserted that the latter "... impossible result<br />
is a necessary consequence of assuming any formula for the potential which<br />
introduces negative terms into the coefficient of v2" 397. Hoppe, an historian<br />
who defended Weber’s assumptions in two books, first in 1884 <strong>and</strong> then in 1927,<br />
asserts that in 1875 Weber succeeded in showing that <strong>with</strong> a proper<br />
reinterpretation of his equation for the vis viva, Helmholtz’s criticism can be<br />
394 Neumann Amp u Web p.337.<br />
395 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen, insbesonder uuber das<br />
Prinzip der Erhaltung der Energie." 1871. In Werke 6 vols. Berlin, 1892-4. Vol.4. Pp. 247-99.<br />
Tr. in Phil Mag 43 (1872): 1-20 <strong>and</strong> 119-49.<br />
396 Whittaker Aether p.204.<br />
397 Maxwell Treatise par.854.
efuted. But again in 1881 Helmholtz criticised Weber on the grounds of a<br />
possible physical case in which the law would have given an imaginary value of<br />
the velocity 398. Hoppe noted other aspects of the controversy: C. Neumann in<br />
1871, 1875 <strong>and</strong> 1877 <strong>and</strong> Zoellner in 1876 published in favour of Weber <strong>and</strong><br />
against Helmholtz. Helmholtz in 1876 recognized that Rowl<strong>and</strong>’s experiments on<br />
convection were not in disagreement <strong>with</strong> Weber’s law. In 1887 Budde made a<br />
very sophisticated analysis of the theories of Weber, Clausius <strong>and</strong> Riemann, <strong>and</strong><br />
planned a series of experiments "in order to prove their validity" 399. C.<br />
Neumann’s contributions of 1877 <strong>and</strong> 1898 are extremely interesting for an<br />
objective judgment of this long dispute. C. Neumann in fact was a defender of<br />
Weber’s position. Still in 1877 he answered Helmholtz’s criticism of 1872<br />
against Weber in the following terms: "The objection that follows (in Helmholtz’s<br />
argument ) does not bear on the usual principle of energy, but on a completely<br />
novel principle which is here pronounced for the first time. While the usual<br />
principle of energy dem<strong>and</strong>s for each system the existence of an energy function,<br />
i.e. the existence of a function which has the quality to increase in each time<br />
interval by the same amount as is the work which is being performed on the<br />
system from the outside - this new principle dem<strong>and</strong>s not only the existence of<br />
such a function, but at the same time a very specific feature of it, in as much as it<br />
asserts that the kinetic part of this function must invariably be positive." 400 In<br />
Weber’s approach the electrokinetic potential has terms containing squared<br />
velocities like the kinetic energy ones. But in contrast <strong>with</strong> st<strong>and</strong>ard kinetic terms<br />
which are multiplied by a positive coefficient like m, these new terms can be<br />
multiplied by negative coefficients, like electric charges. The sum of the resulting<br />
positive <strong>and</strong> negative terms containing squared velocities can be negative. From<br />
Helmholtz’s point of view, this sum represents a negative kinetic energy <strong>and</strong> thus<br />
the new requirement of positive T was postuled . C. Neumann explicitly asserts<br />
that physical principles are not to be considered established once <strong>and</strong> for all.<br />
Modifications are possible <strong>and</strong> the one just introduced by Helmholtz must be<br />
carefully analysed. C. Neumann outlines three shifts in PCE: the principle of<br />
conservation of vis viva, the principle of conservation of energy, <strong>and</strong> Helmholtz’s<br />
new principle of positive kinetic energy. Being a new principle, for Neumann it<br />
cannot be considered a secure ground for debate. Helmholtz’s objections against<br />
398 Hoppe Histoire p.591.<br />
399 Hoppe Histoire p.592.<br />
400 C.Neumann Amp <strong>and</strong> Web p.322.
Weber have to be analysed "apart from that principle". That is, the requirement of<br />
positive kinetic energy as a general rule is denied, but the possibility of specific<br />
inadmissible conclusions from Weber’s law (which admits of negative kinetic<br />
energy) is accepted. But as to this second problem C. Neumann also remarks that<br />
Helmholtz’s criticism can be refuted: "The fact stressed by Helmholtz, i.e. the<br />
occurrence of infinitely great accelerations, leads to either the conclusion that<br />
Weber’s law is inadmissible, or that the singular states are not possible. And it<br />
would therefore be overhasty to jump to the conclusion that one alternative is<br />
preferable over the other <strong>and</strong> to take this fact to speak against Weber’s law" 401.<br />
In fact, in his 1877 analysis, C. Neumann admits the second alternative <strong>and</strong><br />
refuses the first: "Indeed, this fact would only be evidence against Weber’s law if<br />
one could prove, in an example, that those singular states are indeed possible. We<br />
shall see later how far away we are from succeeding to find such proof." 402. But<br />
twenty years later, in 1898, his position was different. He finally published the<br />
second volume on "Electric forces". The first volume had been published in 1873<br />
<strong>and</strong> was dedicated to an analysis of AmpÏere <strong>and</strong> F. Neumann’s theories.<br />
Following the plan of the first volume the second should have dealt <strong>with</strong> Weber’s<br />
<strong>and</strong> Kirchhoff’s theories. But instead the 1898 volume was dedicated to<br />
Helmholtz’s theory. Neumann had finally accepted <strong>Helmholtz's</strong> theory.<br />
It is well known that Helmholtz’s contributions to electrodynamics were also<br />
relevant from the experimental point of view. In fact in the light of the<br />
equivalence of theoretical predictions of the three theories for closed currents, he<br />
stressed the importance of experimenting <strong>with</strong> open currents. This approach was<br />
to lead to Hertz’s experiments of the late eighties, but still in the seventies<br />
Helmholtz suggested three important experiments to Schiller (1874), Rowl<strong>and</strong><br />
(1876) 403 <strong>and</strong> Hertz himself (1879). The first two experiments were interpreted in<br />
favour of the dielectric theories (whether Helmholtz’s or Maxwell’s). Rowl<strong>and</strong> in<br />
particular demonstrated the electrodynamic effects caused by convection<br />
currents. Hertz’s experiment was to deny the relevance of Weber’s electrical<br />
inertia 404.<br />
401 Ibidem p.324.<br />
402 Ibidem p.324.<br />
403 Woodruff "Helmholtz" pp.308-10.<br />
404 Helmholtz n.394 p.6.
But was important here is that despite the fact that, in the late seventies, no<br />
conclusive evidence existed for a specific theory, Helmholtz came gradually to<br />
consider dielectric theory as correct.<br />
"In the Faraday lecture of 1881 he predicted the decline of action-at-a-distance<br />
on the Continent <strong>and</strong> lent full support to Maxwell’s theory" 405.<br />
But he was still convinced, against Maxwell, that electricity consisted ultimately<br />
in discrete charges, "atoms of electricity". Thus the main element accepted from<br />
Maxwell was the role of the dielectric <strong>and</strong>, gradually, contiguous action.<br />
Helmholtz’s shift towards contiguous action <strong>and</strong>, in particular, the relation of the<br />
shift <strong>with</strong> the new version of PCE (Poynting's principle of local conservation)<br />
deserves detailed attention. Helmholtz’s views at the beginning of the eighties<br />
had already undergone some changes. His 1847 Newtonian model of central<br />
forces depending only on positions was changed in 1870 into a model of forces<br />
<strong>and</strong> torques. The theory of the dependence of forces on the positions was also<br />
lost, when Helmholtz accepted the general electrodynamic potential, a concept<br />
first introduced by F.Neumann, Weber <strong>and</strong> Clausius. In 1872 Helmholtz asserted<br />
the further condition on PCE, i.e. that the generalised kinetic energy of the<br />
electrical system (deriving partly from electrodynamic potential) must always be<br />
positive. Very few aspects of Helmholtz’s early conception remained at the<br />
beginning of the eighties. One of these was the distinction between kinetic <strong>and</strong><br />
potential energy. While shifting towards contiguous action, a conception that in<br />
the end would have denied this distinction, Helmholtz underwent a second,<br />
almost contemporary, shift: the Principle of Least Action was now to be<br />
considered the key to physical research, more than the Principle of <strong>Conservation</strong><br />
of <strong>Energy</strong>.<br />
Unification, again ( 1884-94)<br />
In a long paper of 1886, Helmholtz introduces a new unifying<br />
programme in physics, based on a heuristic principle different from energy<br />
conservation : the principle of least action (PLA). The principle of conservation<br />
of energy is ab<strong>and</strong>oned as the main guide in physics :<br />
"it in fact holds in a great variety of cases when least action does not.<br />
The latter is thus more specific".<br />
405 Turner, Steven. "Hermann von Helmholtz." In DSB 6 ,1973. Pp.241-53. P.252.
The real starting point for the 86 paper was the search for an<br />
expression of the kinetic potential in Maxwell's theory. The old restrictive<br />
hypothesis according to which velocity is solely part of the vis viva, as a<br />
homogeneous quadratic function, has to be dropped. Helmholtz, at the very<br />
moment when the contiguous action theory was receiving its own non<br />
mechanistic local principle of conservation, gives up <strong>with</strong> the requirement so<br />
strongly defended for almost forty years: he recognizes that energy can be any<br />
function whatever of the coordinates <strong>and</strong> velocities. The retreat from the 1847<br />
positions is now almost complete: a lasting element is that the new interpretation<br />
of energy is still in the mechanical framework.<br />
But Helmholtz, now sixty-four <strong>and</strong> at the height of an extraordinary<br />
career, is again ready to start working towards new unifying horizons. The new<br />
regulative principle, PLA, should not be confined to mechanics <strong>and</strong> optics;<br />
Helmholtz applied it to electrodynamics <strong>and</strong> attempted to extend its range to<br />
thermodynamics:<br />
"From these facts we may even now draw the conclusion that the<br />
domain of validity of the principle of Least Action has reached far beyond the<br />
boundaries of the mechanics of ponderable bodies. Maupertuis’ high hopes for<br />
the absolute general validity of his principle appear to be approaching their<br />
fulfilment, however slender the mechanical proofs <strong>and</strong> however contradictory the<br />
metaphysical speculations which the author himself could at the time adduce in<br />
support of his new principle. Even at this stage, it can be considered as highly<br />
probable that it is the universal law pertaining to all processes in nature ..... In<br />
any case, the general validity of the Principle of Least Action seems to me<br />
assured, since it may claim a higher place as a heuristic <strong>and</strong> guiding principle in<br />
our endeavour to formulate the laws governing new classes of phenomena." 406<br />
These statements are part of the 1886 paper on PLA. A second paper<br />
was to appear in 1887. C. Neumann analysing Helmholtz’s work in a long<br />
treatise of 1898, remarks that <strong>Helmholtz's</strong> electrical investigations can be divided<br />
into two groups: the first (1870-75) has its roots in the general conceptions of<br />
Newton’s gravitational theory, i.e. in the principle of direct actionat-a-distance.<br />
406 Helmholtz, Hermann. "Ueber die physikalische Bedeutung des Princips der<br />
kleinsten Wirkung." In Crelle's Jour 100 (1886): 137-66 <strong>and</strong> 213-22. Repr. in WA 3. Pp.203-<br />
. Quot. from; Yourgrau <strong>and</strong> M<strong>and</strong>elstam. Variational Principles in Dynamics <strong>and</strong> Quantum<br />
Theory. 2nd ed. London: Pitman, 1960. P.143.
The second (1892-94), follows the investigations of Faraday, Maxwell <strong>and</strong> Hertz<br />
<strong>and</strong> shares its general basic features <strong>with</strong> Fourier’s theory of heat conduction.<br />
"Like the latter it rests on the idea that the true origins of any changes<br />
occurring in any point of the universe must be found in the immediate vicinity of<br />
this point" 407.<br />
It is thus interesting to analyse the reasons for this double almost<br />
contemporary shift: from action-at-adistance to contiguous, from conservation of<br />
energy to least action, <strong>and</strong> their possible connection. The analysis is here outlined<br />
only <strong>with</strong> reference to the modification of the concept of energy <strong>and</strong> to the<br />
expression of PLA in the ’86 paper.<br />
Helmholtz recognizes the following points:<br />
a) H=F-L <strong>and</strong> E=F+L (where H is the Hamilton function, F potential<br />
energy <strong>and</strong> L vis viva); the name of kinetic potential should be attributed to H,<br />
after F. Neumann’s work on mutual potential, <strong>and</strong> Clausius’ work on<br />
electrodynamic potential. As already recalled the old restrictive hypothesis<br />
according to which the velocity is solely part of the vis viva, as a homogeneous<br />
quadratic function, has to be dropped. H can be any function whatsoever of<br />
coordinates <strong>and</strong> velocities.<br />
b) In the electrodynamic case such functions have been given by F.<br />
Neumann, Clausius <strong>and</strong> Weber. But the dielectric has to be taken into account,<br />
following recent results. In this case, the form of the function differs from that for<br />
ponderable masses.<br />
c) The starting point for the research of the ’86 paper was the search<br />
for an expression of the kinetic potential in Maxwell’s theory.<br />
d) In a great variety of cases, the conservation of energy holds, while<br />
least action does not. Thus PLA expresses a particular feature of the conservative<br />
forces not intrinsically connected <strong>with</strong> their conservative character. That is, it<br />
says something else which is more specific 408. In view of these remarks, my<br />
opinion is that Helmholtz’s shift towards Maxwell in the late seventies was due<br />
to the possibility of localising the energy in the dielectric. This allowed for a<br />
better explanation of optics, of experimental results <strong>and</strong> also for a distinction<br />
between kinetic <strong>and</strong> potential energy of the dielectric medium to be maintained,<br />
while in the other theories this distinction was lost. At the same time, most of the<br />
407 Neumann, Carl. Die Elektrischen Kräfte. Part 2. Leipzig: Teubner, 1898. P.IV.<br />
408 Helmholtz "kleinst Wirk" Introduction <strong>and</strong> par.2.
different competing theories showed an agreement <strong>with</strong> some versions of PCE,<br />
mainly <strong>with</strong> the impossibility of perpetuum mobile. That is, they admitted a<br />
potential. Thus, strictly speaking, there was no longer any heuristic power in the<br />
PCE (still a global one in Helmholtz’s view). Either a more specific version of<br />
PCE or a more specific regulative principle was needed. The more specific<br />
version of PCE was to be connected <strong>with</strong> the localisation of energy of Poynting,<br />
but Helmholtz seems not to have understood its relevance, the more specific<br />
principle was in Helmholtz’s view PLA. Moreover the PLA version had to be in<br />
agreement <strong>with</strong> contiguous action: in fact a PLA related <strong>with</strong> Maxwell’s theory<br />
was Helmholtz’s starting point in 1886. At this stage the two shifts had been<br />
accomplished before Hertz’s experiments.<br />
In 1894 Helmholtz publishes a small volume entitled "Introduction to<br />
the Lectures on Theoretical Physics". The first two sections, Introduction <strong>and</strong><br />
Part I, are closely related to the themes discussed fourty years before in the<br />
Erhaltung <strong>and</strong> <strong>with</strong> them we will be concerned here. The third section, Part II, is<br />
very similar to the paper "An Epistemological Analysis of Counting <strong>and</strong><br />
Measurement".<br />
Introduction <strong>and</strong> Part I are of great relevance to focus some aspects of<br />
<strong>Helmholtz's</strong> methodological beliefs. Namely the conditions of possibility of<br />
underst<strong>and</strong>ing nature, the relations betwen force <strong>and</strong> law, the role of regulative<br />
principles in framing our scientific knowledge.<br />
What had been achieved: a turn of the century viewpoint<br />
<strong>Energy</strong> theory changed the shape of physics : after 1847 physical laws<br />
no longer had only to face the challenge of experimental results, but also had to<br />
be judged on more theoretical grounds, in their relation <strong>with</strong> the conservation<br />
principle. But in turn the principle had, from the beginning, different<br />
formulations.<br />
The growth of these formulations was joined <strong>with</strong> the growth of<br />
alternative research programs in the second half of the century. The energetist<br />
movement took off, <strong>with</strong> its attempts to overthrow the mechanical worldview,
while the electromagnetic <strong>and</strong> the statistical approach, both deeply connected<br />
<strong>with</strong> energy problems, were actively pursued. Since energy conservation became<br />
one of the basic chapters of physics, the champions of the different research<br />
programs dedicated a great deal of logical <strong>and</strong> historical analysis to an<br />
underst<strong>and</strong>ing <strong>and</strong> framing of the various contributions <strong>and</strong> developments.<br />
Helmholtz was a leader in this spread of theoretical physics in the second half of<br />
the century, <strong>and</strong> is interesting to find out what was the judgement of his fellow<br />
scientists on his approach to PCE.<br />
I have already dealt <strong>with</strong> the first debate on the history <strong>and</strong> role of the<br />
energy pioneers, but besides Tyndall (1863) <strong>and</strong> Tait (1868 <strong>and</strong> 1876), from the<br />
seventies a number of books appeared on the history <strong>and</strong> foundations of energy<br />
theory which, while demonstrating the now recognized importance of the subject,<br />
gave interesting assessments of <strong>Helmholtz's</strong> contributions. Among them 409:<br />
Maxwell (1870 <strong>and</strong> 1877), B.Steward (1874), Stallo (1882), Lodge (1929), <strong>and</strong><br />
in Germany : Mach (1872, 1883 <strong>and</strong> 1896), Planck(1887), Helm (1887 <strong>and</strong><br />
1898), Ostwald (1903 (tr fr.1912) <strong>and</strong> 1908), Haas (1909), <strong>and</strong> in France :<br />
Duhem (1895,1905), Poincarè (1892 <strong>and</strong> 1902). Rowl<strong>and</strong> (1882) analysis of the<br />
experimental determinations of the mechanical equivalent of heat is also<br />
interesting.<br />
Philosophers too at the beginning of the century took a great interest in<br />
evaluating the role of <strong>Helmholtz's</strong> formulation of energy conservation, among<br />
these : Meyerson (1907) <strong>and</strong> Cassirer (1910). Important among the historians is<br />
the contribution of Mertz (1965 rep), that anticipates some modern<br />
historiographical claims.<br />
I will confine myself here at discussing some remarks on <strong>Helmholtz's</strong> energy<br />
theory <strong>and</strong> on his specific methods made by Planck, Helm <strong>and</strong> Poincarè.<br />
Planck outlines a basic problem in the formulation <strong>and</strong> application of the<br />
principle of conservation of energy, in whichever form : it is impossible to find<br />
the primary expression of the energy 410; the substantialization of energy is open to<br />
a certain degree of arbitrariness 411; there is a difficult theory/ experiment<br />
interplay 412. These general remarks are valid for all the expressions of the<br />
principle. What are the merits <strong>and</strong> demerits of <strong>Helmholtz's</strong> specific formulation?<br />
409 See n.7.<br />
410 Planck Prinzip p.114.<br />
411 Planck Prinzip p.104.<br />
412 Planck Prinzip Pp.45-47 e 235-40.
Planck asserts that the reduction of all the forms of energy to two main ones is a<br />
great merit, a simplification <strong>and</strong> a great heuristic tool, but that nevertheless this is<br />
insufficient to deal <strong>with</strong> in complicated situations, like electromagnetism, still<br />
partly unknown. This means that the principle 'per se', <strong>with</strong>out the contribution of<br />
experimental results, cannot give definite answers.<br />
To apply <strong>Helmholtz's</strong>'s principle to a process that takes place in a system of<br />
bodies it is necessary to add together all the different kinds of variations of vis<br />
viva on one side, the sum of tension forces on the other side <strong>and</strong> equate the two<br />
groups. The sum of all the identified terms representing the vis viva <strong>and</strong> the<br />
tension forces at a given instant is the "force" (energy) of the system, constant if<br />
the system is isolated. Of course all the identified different terms of the sum have<br />
to be numerically expressed on the basis of a common unity of measurement,<br />
usually mechanical work; thus their mechanical equivalent has to be found. But in<br />
this process of identification of the different terms contributing to the energy of<br />
the system we find a methodological difficulty:<br />
"there is not a general rule <strong>with</strong> which we can calculate a priori the value of the<br />
equivalent (i.e. the expression of the different terms), independently of the<br />
principle itself" 413<br />
Often in the past wrong identifications of the equivalents led to wrong<br />
conclusions (Planck quotes Descartes <strong>and</strong> Carnot 414).<br />
<strong>Helmholtz's</strong> principle <strong>with</strong> its reduction of all the energy terms to two main forms,<br />
kinetic <strong>and</strong> positional, is <strong>with</strong>out doubt a great heuristic tool to find out the<br />
different energy terms, but does not solve all the problems. That is, there is<br />
always an indetermination <strong>and</strong> then an arbitrariness in the identification <strong>and</strong><br />
expression of the energy terms.<br />
There is in fact a complicated interplay between the general framework of the<br />
principle <strong>and</strong> the empirical laws.<br />
From an existing <strong>and</strong> accepted empirical law the expressions of the tension<br />
forces <strong>and</strong> of the vis viva have to be identified (<strong>and</strong> the law rededuced). Instead<br />
for a realm of phenomena not yet described by laws or <strong>with</strong> laws not sufficiently<br />
corroborated, if we assume the principle to hold true some useful guide-lines for<br />
the formulations of empirical laws can be produced. Sometimes the principle<br />
leads to discover new laws, in other cases the discovery of new experimental<br />
laws will add new energy terms to the principle. Again new experimental data<br />
413 Planck Prinzip p.38<br />
414 Planck Prinzip Pp.7-9 e 13-17
can modify its specific theoric formulation, but not its validity, that Planck<br />
considers true as far as numerical results are concerned 415. Following Planck in<br />
fact the numerical value of the work equivalent of a transformation of a system<br />
between two states is always the same, independently of the way in which the<br />
transformation takes place. This he considers as an experimental result. But the<br />
theoretical interpretation of the energy terms can be achieved in different ways in<br />
different theories: the process of "substantialization" is open to a certain<br />
arbitrariness, always useful for the progress of knowledge 416. Planck ends his<br />
1887 analysis pointing to one of this progress, the overcome of the action at a<br />
distance approach in electrodynamics, based on the (theoretical) success of the<br />
new version of the principle of conservation: the local conservation of energy.<br />
There is not a unique way to apply the general framework <strong>and</strong> thus a great part of<br />
the success relies in the ability of the scientist to use this new powerful<br />
theoretical tool: thus <strong>Helmholtz's</strong> 1847 success is due not only to his theoretical<br />
formulation but to his widespread <strong>and</strong> deep knowledge of the most various<br />
branches of natural science, a fact often overlooked by commentators. Only when<br />
his empirical knowledge fails, as in the case of electromagnetism, the<br />
methodological problems of the theory/experiment interplay in the formulation<br />
<strong>and</strong> application of the principle become evident.<br />
So far Planck's warnings. A different approach is outlined by Helm: of the two<br />
roots of <strong>Helmholtz's</strong> principle he accepts the first one (impossibility of perpetual<br />
motion) <strong>and</strong> denies the validity of the second (central forces). Thus Helm, while<br />
accepting a principle of conservation <strong>and</strong> a concept of energy, refuses as<br />
unnecessary <strong>Helmholtz's</strong> two main energy forms: tension <strong>and</strong> living forces. For<br />
Helm there is no theoretical nor practical reason to specify in an energy balance<br />
which is a tension force <strong>and</strong> which a living one, given that what matters is the<br />
work equivalent of a specific energy term. Helm denies 417 both the central forces<br />
hypothesis <strong>and</strong> the mechanical view of nature. <strong>Energy</strong> conservation means energy<br />
correlation <strong>and</strong> the equations are nothing else that sums of work equivalents. The<br />
mechanical view of nature, Helm asserts in line <strong>with</strong> Mayer's approach, does not<br />
add anything to the constancy of the sum of the work equivalents. I want to<br />
underline that the refusal of the privileged status of the central forces does not<br />
necessarily imply the rejection of the mechanical view: Clausius, as discussed<br />
415 Planck Prinzip p.99, see also the Introduction to the 1887 edition.<br />
416 Planck Prinzip Pp.104-5.<br />
417 Helm Energetik p. 41.
above, accepted the mechanical view <strong>and</strong> rejected the limitation to central<br />
forces <strong>and</strong> thus the distinction between positional <strong>and</strong> kinetic energy terms. For<br />
Clausius the specific condition needed to fulfill energy conservation is that the<br />
work done by the forces has to be a total differential 418.<br />
Much different the position of another great scientist: Poincarè. Helmholtz clear<br />
distinction between potential <strong>and</strong> kinetic energy is for Poincarè basic to give a<br />
real meaning to the general concept of energy. From a mathematical point of view<br />
there are many invariants in a transformation <strong>and</strong> <strong>with</strong>out <strong>Helmholtz's</strong> criteria<br />
there would be serious difficulties in making a choice. Thus Poincarè 419 accepts<br />
completely <strong>Helmholtz's</strong> approach <strong>and</strong> believes that this is the only solution to<br />
solve the difficulties connected <strong>with</strong> the arbitrariness of the energy concept.<br />
In 1902 Poincarè published a synthesis of his works in La Science et<br />
l'Hypothèse , a famous book, important here because it shows that energy<br />
considerations were still fundamental for the scientific debate. The first edition of<br />
the first volume of Poincarè’s Electricitè et Optique was published in 1890. It<br />
collects the lectures given between March <strong>and</strong> June 1888 (<strong>and</strong> not 1889 as<br />
erroneously printed on the volume). The whole book concerns Maxwell’s theory.<br />
Poincarè’s Introduction became very famous, for he analyses the difficulties a<br />
French reader has to face reading Maxwell’s Treatise . The difficulties are<br />
related to a lack of precision <strong>and</strong> logical order, a lack which conceals Maxwell’s<br />
main result. This main result plays a great part in Poincarè’s analysis of the CET<br />
debate. Which is Maxwell’s main result?<br />
"Pour dèmontrer la possibilitè d’une explication mècanique de l’èlectricitè, nous<br />
n’avons pas à nous prèoccuper de trouver cette explication elle-meme, il nous<br />
suffit de connaitre l’expression de deux fonctions T et U qui sont les deux parties<br />
de l’ènergie, de former avec ces deux fonctions les equations de Lagrange et de<br />
comparer ensuite ces èquations avec les lois expèrimentales." 420<br />
Several points are contained in this statement: the necessity for a mechanical<br />
explanation of electricity, the sufficiency of the possibility of the explanation, the<br />
role of a PCE <strong>with</strong> a sharp distinction between T <strong>and</strong> U, the link of a Lagrangian<br />
derivation <strong>with</strong> this PCE. The actual kind of mechanical explanation is not<br />
1968.<br />
418 See above.<br />
419 Poincaré, Henry. La Science et l'Hypothèse . Paris, 1902. Rep Paris: Flammarion,<br />
420 Poincarè, Henry. Electricité et Optique. I Les Theories de Maxwell. Paris: Carré,<br />
1890; pp.XIV-XV.
considered relevant. In Poincarè’s view, the demonstration of the possibility of a<br />
mechanical explanation i.e. the fulfillment of Lagrange equations, is the main<br />
point in Maxwell’s analysis. This allows him to accept the difficulties <strong>and</strong><br />
contradictions that sometimes blemish the British masterpiece. In my analysis,<br />
Poincarè’s remarks are fundamental for two reasons: first, because they show the<br />
importance of PCE <strong>and</strong> PLA, i.e. of regulative principles, as grounds of debate;<br />
second, because they show that Maxwell’s merit was considered to be the<br />
application of <strong>Helmholtz's</strong> PCE, a specific PCE <strong>with</strong> a sharp distinction of T <strong>and</strong><br />
U. Thus Poincarè is not concerned, unlike Planck, <strong>with</strong> the local conservation in<br />
Poynting’s sense, but still judges Maxwell's theory on the basis of <strong>Helmholtz's</strong><br />
PCE, the same that Maxwell adopts. For Poincarè in 1890, the sharp distinction<br />
between T <strong>and</strong> U is still very important, <strong>and</strong> we are going to see that he will<br />
consider it basic in later works too. But still this is not Poincarè only meaning of<br />
conservation: in 1890 he asserts that a consequence of the assumed conservation<br />
of energy is the existence of a force function (= - U, where U is the potential<br />
energy) so that the equations of movement can be expressed as 421:<br />
etc. Sometimes Poincarè asserts that one of the expressions of conservation of<br />
energy is the existence of a potential depending only on positions 422; in the case<br />
of two electric circuits the terms expressing an exchange of energy are:<br />
where is the electric energy provided by two batteries;<br />
is the Joulean heat produced in the circuit <strong>and</strong> dT is the part of the variation of<br />
the electrodynamic potential in itself of the system of the two circuits due to the<br />
displacement of the circuits. Now PCE asserts that the previous expression has to<br />
be zero for a closed cycle or to be an exact differential in other cases 423. In this<br />
last example, reference to potential <strong>and</strong> kinetic energy in the expression of PCE is<br />
avoided. Other relevant passages of this first edition are the statements of the<br />
existence of two electrostatic theories deducible from Maxwell’s <strong>and</strong> that in the<br />
first theory electrostatic energy cannot be considered as potential 424. Instead<br />
421 Poincarè El et Opt p.X.<br />
422 Poincarè El et Opt p.117 (par.106).<br />
423 Poincarè El et Opt p. 165 (par.149).<br />
424 Poincarè El et Opt p.92 (par.84).
Poincarè agrees that the electrodynamic potential of a system of currents is the<br />
kinetic energy of the ether 425.<br />
In contrast to the great interest in energy theories, little room is left for Hertz’s<br />
experiments.<br />
Analysing Weber’s theory, Poincarè asserts that it agrees <strong>with</strong> PCE, as far as the<br />
work of electrodynamic repulsion is equal to the differential - dc of the potential<br />
c. 426 In this case Poincarè is using the potential law as PCE, but somewhere else<br />
he adopts Helmholtz’s PCE, <strong>and</strong> clearly refers to the distinction between kinetic<br />
<strong>and</strong> potential energy. In the analysis of Helmholtz’s theory the expression of PCE<br />
is the following: the variation of T+U must be equal to the work performed by the<br />
external electromotive forces (chemical, thermoelectrical, etc.) minus the Joulean<br />
heat 427. The need for the distinction of T <strong>and</strong> U appears relevant in the subsequent<br />
analysis of the different theories: Weber’s is rejected, despite its fulfilment of<br />
PCE, because of the negative value of the constant K in Helmholtz’s general<br />
potential. In fact K < O leads to negative kinetic energy <strong>and</strong> instabilities 428. The<br />
relevance for Poincarè of PCE <strong>and</strong>, moreover, his specific preference for a PCE<br />
involving T <strong>and</strong> U has already been outlined. His preference for Maxwell was a<br />
result of this choice. Equally remarkable is that Poincarè’s made a link between<br />
PCE (<strong>with</strong> T <strong>and</strong> U sharply divided) <strong>and</strong> the Lagrangian derivation: he never<br />
considered the possibility of a Lagrangian derivation <strong>with</strong> an electrokinetic<br />
potential, <strong>and</strong> thus refers the possibility of a mechanical explanation (given by the<br />
Lagrangian derivation) to the possibility of dividing T <strong>and</strong> U.<br />
Poincarè’s ideas on energy are clarified in a work of 1892 429. The principle of<br />
conservation is called here Mayer’s principle <strong>and</strong> Poincarè wonders at the<br />
success it has among other physical laws. A reason cannot be found in its<br />
connections <strong>with</strong> the impossibility of perpetual motion (from which it can be<br />
derived only in the case of reversible phenomena). Moreover the attempts at a<br />
clear definition are useless: "it is impossible to find a general definition of it". The<br />
principle disappears when generalised <strong>and</strong> applied to the universe. The only<br />
425 Poincarè El et Opt p.169 (par.152).<br />
426 Poincarè, Henry. Electricité et Optique. II. Les Theories de Helmholtz. Paris:<br />
Carré, 1891; p.31, 41 (par.18-20).<br />
427 Poincarè El et Opt 2 p.69 (par.31).<br />
428 Poincarè El et Opt 2 p.75 (par.34).<br />
429 Poincarè, Henry. Thermodynamique. Paris: Carré, 1892; p.IX. Repr. in Poincarè<br />
La Science pp.144-9.
meaning left is: "There is something which is constant". But what is this<br />
something? Poincarè distinguishes two cases: a universe whose evolution is<br />
completely determined by the values of n parameters <strong>and</strong> of their derivatives <strong>and</strong><br />
a system in which there are p of the n parameters that vary independently (that is,<br />
the system is a limited one, interacting <strong>with</strong> the exterior world). In the first case,<br />
the n existing differential equations admit n-1 first integrals, i.e. constants. It<br />
would be difficult to decide which deserves the name energy. Thus the principle<br />
is nonsense. In the second case, the principle expresses a limitation: the n-p<br />
relations admit a combination whose first member is a complete differential; if the<br />
work of the external forces <strong>and</strong> the heat exchanged is zero, the integral of the<br />
resulting zero differential is a constant, i.e. the energy. 430 Here conservation of<br />
energy as existence of a complete differential, is clearly defined, but <strong>with</strong> the<br />
constraint of admitting a limited system interacting <strong>with</strong> the outside. At this stage,<br />
Poincarè does not discuss the Helmholtzian interpretation of energy as the sum of<br />
T <strong>and</strong> U, but it will be seen that this was to be his choice, in order to avoid the<br />
difficulties just expressed.<br />
In 1902 Poincarè improves his analysis: now again, as in 1890, the distinction<br />
between T <strong>and</strong> U is fundamental to give a specific meaning to energy. In an<br />
analysis that became famous, Poincarè showed that, to give a specific meaning to<br />
energy, its terms have to be of a particular form, one depending on square<br />
velocities <strong>and</strong> one on positions 431. Weber’s case is explicitly quoted: a distinction<br />
between kinetic <strong>and</strong> potential energy being impossible, how can we define<br />
energy? 432 In this analysis, where PCE is now called Helmholtz’s <strong>and</strong> not<br />
Mayer’s principle, the cases in which energy cannot clearly be divided into T <strong>and</strong><br />
U, (when for instance also an internal energy Q appears, not clearly distinguished<br />
from the other two), are considered tautological expressions of conservation:<br />
something is conserved, but the whole assertion is untestable due to its<br />
unspecificity. Poincarè is thus led to what Planck called a substantialisation, <strong>and</strong><br />
the substantialisation is clearly expressed in the electromagnetic case. First of all<br />
the distinction between T <strong>and</strong> U. Second their interpretation in electromagnetic<br />
terms <strong>and</strong> their localisation:<br />
"Et alors Maxwell s’est dem<strong>and</strong>è s’il pouvait faire ce choix et celui des deux<br />
ènergies T et U, de facon que les phènomenes èlectriques satisfassent à ce<br />
430 Poincarè Thermod p.XI. Repr. in Poincarè La Science p.147.<br />
431 Poincarè La Science pp.139-44.<br />
432 Poincarè La Science p.141.
principe. L’expèrience nous montre que l’ènergie d’un champ èlectromagnètique<br />
se dècompose en deux parties, l’ènergie èlectrostatique et l’ènergie<br />
èlectrodynamique. Maxwell a reconnu (que si l’on regarde) la premiére come<br />
reprèsentant l’ènergie potentielle U, la seconde comme reprèsentant l’ènergie<br />
cinètique T." 433<br />
Maxwell thus solves the problem, allowing in addition a Lagrangian derivation of<br />
the equations <strong>and</strong> so fulfills the possibility of a mechanical explanation of<br />
electromagnetism. The grounds of acceptance of Maxwell’s theory in Poincarè’s<br />
analysis are clearly the ones referring to energy. Maxwell’s (<strong>and</strong> not Poynting’s)<br />
approach is preferred in view of its links <strong>with</strong> the mechanical expression of both<br />
PCE <strong>and</strong> PLA. This judgment is explicit already in 1890 <strong>and</strong> was to be reasserted<br />
in 1902 434. Hertz’s experiments, instead, up to 1894 were considered doubtful<br />
<strong>and</strong> in 1902 were considered to lend only indirect support. The<br />
desubstantialisation of ether theory was to weaken the experimental difference<br />
between contiguous action <strong>and</strong> delayed action-at-a-distance theories, but was to<br />
strengthen the theoretical difference between the PCE’s of the latter theories <strong>and</strong><br />
a local PCE expressed by the contiguous action theory. Poincarè’s choice of<br />
Maxwell from 1890 to 1902 is not referred to Poynting’s development (second<br />
step of localisation) but to Maxwell’s substantialisation (first step). This is due to<br />
the relevance he attributed to the asserted possibility of a mechanical<br />
explanation 435. Poincarè’s approach is very different from Hertz’s axiomatic<br />
assumption of Maxwell’s equations <strong>and</strong> his rejection of both a mechanical<br />
explanation <strong>and</strong> of a Lagrangian derivation. But still PCE, <strong>and</strong> specifically<br />
<strong>Helmholtz's</strong> PCE, was the principal reason for Poincarè’s choice of Maxwell.<br />
In the important process of the emergence of theoretical physics<br />
discussions on the meaning <strong>and</strong> the role of <strong>Helmholtz's</strong> approach to energy<br />
conservation were basic. Acceptance or denial of the mechanical world view, of<br />
the sharp distinction between kinetic <strong>and</strong> potential energy <strong>and</strong> of the privileged<br />
status of the central forces depending only on the distance were among the key<br />
issues, while others, non strictly mechanical interpretations of the principle of<br />
conservation, started to become mere <strong>and</strong> more important. At the end of the<br />
century in every research programme (mechanical, electromagnetic, energetic,<br />
thermodynamic) a specific version of the principle was put forward, <strong>and</strong> new<br />
433 Poincarè La Science p. 223.<br />
434 Poincarè La Science pp.216-25.<br />
435 Poincarè La Science pp.216-25.
interpretations at the beginning of the new century were to be proposed in the<br />
new programmes of relativity <strong>and</strong> quantum physics. The theory-experiment<br />
interplay did not enjoy any longer a privileged position: the theory-principle one<br />
was now at the forefront of research, showing the lasting importance of<br />
<strong>Helmholtz's</strong> 1847 methodology. It is significant that the emergence of theoretical<br />
physics was joined <strong>with</strong> a deep work on the history of physics, <strong>with</strong> approaches<br />
that often were methodologically sophisticated, even for today's st<strong>and</strong>ard.<br />
Contemporary Historiography<br />
New trends in energy studies cannot avoid discussing what <strong>with</strong>out doubt<br />
started the recalled fourth phase in the debate on the history of energy<br />
conservation: Thomas Kuhn's paper on the "simultaneous discovery" of "energy<br />
conservation" 436, published in 1959. Kuhn's paper is still unanimously defined as<br />
challenging, for the lack in contemporary historiography of a rival synthesis 437.<br />
Nevertheless analysing <strong>Helmholtz's</strong> Erhaltung , <strong>and</strong> the related primary <strong>and</strong><br />
436 Kuhn, Thomas. "<strong>Energy</strong> <strong>Conservation</strong> as an Example of Simultaneous Discovery."<br />
In Critical Problems in the History of Science . M.Clagett ed. Madison: Wisconsin U.P.,<br />
1959. Pp. 321-356. Rep. in Kuhn, Thomas. The Essential Tension, Chicago: Chicago U.P,<br />
1977. Pp. 66-104.<br />
437 Cantor, Geoffrey. "Locating the First Law of Thermodynamics." In New<br />
Perspectives in Nineteenth-Century Science. University of Kent, 1984.
secundary literature, I started doubting of the correctness of some of Kuhn's main<br />
historical <strong>and</strong> historiographical claims.<br />
Six not minor points are, in my view, at issue: 1) that in 1845-46<br />
"Helmholtz fails to notice that body heat may be expended in mechanical<br />
work" 438; 2) that the concept of work of the old mechanical engineering tradition<br />
was "all that which is required" <strong>and</strong> "the most decisive contribution to energy<br />
conservation made by the nineteenth-century concern <strong>with</strong> engines." 439; 3) that<br />
"Helmholtz was not, however, aware of the French theoretical engineering<br />
tradition. Like Mayer he derives the factor of1/2 in the definition of energy of<br />
motion <strong>and</strong> is unaware of any precedent for it." 440; 4) that Helmholtz "fails<br />
completely to identify as work or Arbeitskraft <strong>and</strong> instead calls it the "sum<br />
of the tensions.." 441; 5) that "the dominance of contact theory in Germany" might<br />
"account for the rather surprising way in which both Mayer <strong>and</strong> Helmholtz<br />
neglect the battery in their accounts of energy transformations" 442; 6) that<br />
"Helmholtz was able by 1881 to recognise important Kantian residues " in the<br />
Erhaltung "that had escaped his earlier censorship" 443 <strong>and</strong> that this is evidence of<br />
influences of Naturphilosophie .<br />
I will show below that: 1)Helmholtz before 1847 was aware of the heatwork<br />
interconvertibility, despite lacking a numerical equivalent; 2) the concept of<br />
work was not a precious contribution to energy conservation till it acquired the<br />
requirements of being a total differentiaL; 3) Helmholtz was well aware of the<br />
French tradition; he utilised but did not rederive the new expression of the vis<br />
viva; 4) Helmholtz consciously dropped also the new interpretation of the term<br />
"Arbeit" in favour of "Spannkraft". This was one of his most relevant<br />
contributions. 5) Helmholtz not only showed that contact theory was not against<br />
conservation, but also dedicated the longest section of the Erhaltung to an<br />
analysis of the batteries; 6) Helmholtz did not censor but reinstated in 1847 the<br />
philosophical introduction to the Erhaltung before publication were Kantian<br />
transcendentalism <strong>and</strong> not Naturphilosophie idealism played a main role. In 1881,<br />
438 Kuhn Sim Disc p.95 n. 68.<br />
439 Kuhn Sim Disc p.84 <strong>and</strong> 90.<br />
440 Kuhn Sim Disc p.88.<br />
441 Kuhn Sim Disc p.88.<br />
442 Kuhn Sim Disc p.73<br />
443 Kuhn Sim Disc pp.100-1.
while fighting metaphisicians, he was still stressing Kantian aspects of his earlier<br />
approach.<br />
What is more important some major aspects of <strong>Helmholtz's</strong> Erhaltung<br />
escaped Kuhn's (<strong>and</strong> other historians') attention: 1) <strong>Helmholtz's</strong> methodological<br />
four level structure; 2) his demarcation between theoretical <strong>and</strong> experimental<br />
physics; 3) his overcoming of both the engineering <strong>and</strong> the mathematical<br />
approach to the work concept; 4) the lack of an experimental determination of a<br />
work equivalent of heat <strong>and</strong> the mistranslation of Joule's values; 5) the difficult<br />
(<strong>and</strong> sometimes wrong) theory-experiment interplay in the application of the<br />
principle; 6) the formulation of a lasting methodology <strong>and</strong> of a non lasting<br />
conceptual model of energy.<br />
All that in my view leads to a historiographical problem: in fact if history<br />
does not get clarified "perhaps.. the wrong.. questions have been asked" 444<br />
A closer look at Kuhn's paper is needed: Kuhn claimed that twelve<br />
scientists, divided into three groups of four 445 , between 1832 <strong>and</strong> 1854<br />
"grasped for themselves" essential "elements" of the concept of energy <strong>and</strong> of its<br />
conservation. The paper addresses the problem of why these "elements" became<br />
accessible in those two decades <strong>and</strong> seeks to identify not all the "prerequisites"<br />
which resulted in the theory of energy conservation but only the "trigger factors"<br />
specific to the period. Kuhn outlines three: the "availability of the conversion<br />
processes", the "concern <strong>with</strong> engines" <strong>and</strong> the "philosophy of nature" 446.<br />
In my view two major historiographical problems are related to Kuhn's<br />
use of the terms "simultaneous discovery" <strong>and</strong> "energy conservation" 447. The first<br />
444 Cantor, Geoffrey. "William Robert Grove, the Correlation of Forces <strong>and</strong> the<br />
<strong>Conservation</strong> of <strong>Energy</strong>." In Centaurus 19 (1976): 273-90. P.287; see also P.273.<br />
445 a) combining generality of formulation <strong>with</strong> concrete quantitative applications,<br />
between 1842 <strong>and</strong> 1847: Mayer, Joule, Colding <strong>and</strong> Helmholtz; b) asserting the<br />
quantitative interchangeability of heat <strong>and</strong> work <strong>and</strong> computing a value for the<br />
conversion coefficient, between "before" 1832 <strong>and</strong> 1854: Carnot, Séguin (1839),<br />
Holtzmann (1845) <strong>and</strong> Hirn (1854); c) believing in a single force that in all its<br />
transformations can never be created or destroyed, between 1837 <strong>and</strong> 1844:<br />
Mohr, Grove, Faraday, Liebig. Kuhn Sim Disc pp. 66-9.<br />
446 Ibid p.73<br />
447"energy conservation" is supposed to be the most "striking instance" of a<br />
simultaneous discovery. Kuhn Sim Disc p.69.
term derives from a sociological debate 448 challenging the "priority" approach 449.<br />
Kuhn, adopting this sociological perspective, is obliged to identify different<br />
statements of the pioneers 450 <strong>with</strong> a "simultaneous discovery", despite some<br />
explicit difficulties <strong>with</strong> a stricter historical perspective 451. Paradoxically he gives,<br />
against his own basic historiographical claims 452, a Whiggish explanation 453.<br />
Whatsmore in so doing, <strong>and</strong> here we deal <strong>with</strong> the second historiographical<br />
problem, he adopts a concept of energy that has substantialistic tones 454 <strong>and</strong> that<br />
is by no means shared contemporary physical knowledge 455. Nowhere in the<br />
paper can be found an explicit definition of "energy conservation", a rather<br />
serious weakness, shared by the great majority of the contemporary secondary<br />
448 "the main objective" of the paper is the preliminary identification of the sources of<br />
this phenomenon. Kuhn Sim Disc p.70. Elkana, Yehuda. "The <strong>Conservation</strong> of <strong>Energy</strong>: A case<br />
of Simultaneous Discovery?" In Arch Int Hist Sci 24 (1970): 31-60. Pp 36-7.<br />
449 Kuhn Sim Disc n.8, p.72. See also: Cantor "Locating" p. 1.<br />
450 "no two of our men even said the same thing". Kuhn Sim Disc p.70.<br />
451 "Even to the historian acquainted <strong>with</strong> the concepts of energy conservation, the<br />
pioneers do not all communicate the same thing.", <strong>and</strong> " What we see in their works is not<br />
really the simultaneous discovery of energy conservation". Kuhn Sim Disc p.72.<br />
452 Against the scientist-historians who "retrieved the current contents of the specialty<br />
from past texts of a variety of heterogeneous fields, not noticing that the tradition they<br />
constructed in the process had never existed. In addition, they usually treated concepts <strong>and</strong><br />
theories of the past as imperfect approximations to those in current use, thus disguising both<br />
the structure <strong>and</strong> integrity of past scientific traditions." Kuhn, Thomas. "The Relations<br />
between History <strong>and</strong> the History of Science." In Daedalus 100 (1971) : 271-304. Rep. in Kuhn<br />
Ess Tension. Pp. 127-161. P.149. Kuhn himself warns against the "inappropriateness of our<br />
concept of discovery", as remembered by Cantor Locating p.2. See Kuhn, Thomas. "The<br />
Historical Structure of Scientific Discovery." In Science 136 (1962) :760-64. Rep. in Kuhn Ess<br />
Tension. Pp 165-77.<br />
453 "Only in view of what happened later can we say that all these partial statements<br />
even deal <strong>with</strong> the same aspect of nature". Kuhn Sim Disc p.70.<br />
454 "We know why these elements were there: <strong>Energy</strong> is conserved; nature behaves<br />
that way". Ibid. p.72.<br />
455 "It is important to realize that in Physics today, we have no knowledge of what<br />
energy is.". Feynman, Richard. The Feynman's Lectures on Physics. 3 vols. Addison Wesley,<br />
1963. Vol.1, chapt.4, par 4.1, p.4.2.
literature 456. There is an intrinsic difficulty in assuming an implicit definition of<br />
energy conservation as a historiographical tool: there were many formulations at<br />
the mid of last century, many competing views in the debates of the following<br />
decades, <strong>and</strong> even competing views in this century on the general validity of the<br />
principle 457. Thus what is needed for an historical analysis is an explicit point of<br />
view that does not offer "the formulation" but a possibility of grouping <strong>and</strong><br />
comparing the various expressions. An important contribution was already given<br />
in 1887 in one of the "classics" mentioned above; Planck in fact clarified the<br />
difficult meaning of the principle of energy conservation: following W.Thomson<br />
he defined "energy" as the amount of work that can be done between two states<br />
of a system, he asserted that can be experimentally proved that this (quantity of)<br />
energy is conserved, he outlined a certain arbitrariety in the theoretical expression<br />
of this quantity, he warned 458 about the impossibility of a primary definition of<br />
energy <strong>and</strong> about the everlasting theoretical-empirical interplay that leads to<br />
different formulations <strong>and</strong> applications of the principle. If we accept that "energy"<br />
is not a thing, nor a theory but that it is a relational term 459 whose principle of<br />
conservation, if adopted, had <strong>and</strong> has a number of different formulations we<br />
underst<strong>and</strong> the reason for the difficulty in the historians' works to offer "one"<br />
explicit definition: there are in fact many. Contemporary textbooks, adopting a<br />
superposition principle, list one for every field of inquiry (mechanical,<br />
electromagnetic, nuclear,...) but for each of these at the time of the original<br />
456 When a definition is referred to, there is always a lack of generality; see for<br />
instance: Heimann, Peter. "Conversion of Forces <strong>and</strong> the <strong>Conservation</strong> of <strong>Energy</strong>." In<br />
Centaurus 18 (1974):147-61, p.148 refers to Rankine 1853. Hiebert holds a factorisation<br />
approach: Hiebert, Erwin. "Commentary on the Papers of Thomas Kuhn <strong>and</strong> I.Bernard<br />
Cohen." In Critical Problems in the History of Science . M.Clagett ed. Madison: Wisconsin<br />
U.P, 1959: 391-400. P.392; see also: Hiebert, E. Historical Roots of the Principle of<br />
<strong>Conservation</strong> of <strong>Energy</strong> University of Wisconsin:1962. Rep: New York: Arno Press, 1981.<br />
Pp. 1-6 <strong>and</strong> the related criticisms of Jammer, Max . "Factorisation of <strong>Energy</strong>". In BJPS 14<br />
(1963-4): 160-6.<br />
457 Pauli, Wolfgang. Aufsätze und Vorträge über Physik und Erknenntnistheorie<br />
(1933-58) . Chapt.16.<br />
the "classics".<br />
458 Planck Princip Pp. 104-115.<br />
459 Cantor Locating p.2, recalls Cassirer, showing the relevance of the inheritance of
formulation there were a number competing 460. Thus it is difficult to apply for<br />
what happened at the middle of last century the historiographical categories of<br />
"simultaneous discovery" 461 of "energy conservation".<br />
As a consequence Kuhn's three "trigger" factors have to be discussed as<br />
well. Dealing <strong>with</strong> the first factor, "the concern <strong>with</strong> engines", Kuhn focusses on<br />
the concept of work <strong>and</strong> on three traditions related to it: the older engineering<br />
practice 462, the analytical tradition 463, dealing <strong>with</strong> vis viva conservation, where<br />
the stress was on what was later to be called the potential function, a theoretical<br />
engineering tradition starting <strong>with</strong> Lazare Carnot 464. Kuhn discards the last two 465<br />
<strong>and</strong> asserts that the older tradition is the one that was really effective 466. This<br />
460 For different versions of the principle in the history of electromagnetism see<br />
Bevilacqua, Fabio. The Principle of <strong>Conservation</strong> of <strong>Energy</strong> <strong>and</strong> the History of Classical<br />
Electromagnetic Theory. Pavia: La Goliardica Pavese, 1983.<br />
461Winters criticises the same passage of Kuhn but <strong>with</strong> a different example: the<br />
evolution of <strong>Helmholtz's</strong> ideas on energy shows that there was not a " conservation of energy"<br />
to be discovered. Winters, Stephen. "Hermann von <strong>Helmholtz's</strong> Discovery of Force<br />
<strong>Conservation</strong>." Dissertation. The John Hopkins University, 1985. Pp. 11-12.<br />
462 "a century of engineering practice where its use had been quite independent of<br />
both vis viva <strong>and</strong> its conservation". Kuhn Sim Disc P. 84.<br />
463 Here the concept of work was not recognized as a conceptual entity: "the integral<br />
of a force times differential path elements occurs only in the derivation of the {vis viva}<br />
conservation law. The law itself equates vis viva <strong>with</strong> a function of position coordinates". Ibid<br />
P.86.<br />
464 Here work is the "fundamental conceptual parameter". Ibid. P.87.<br />
465 For the analytical tradition see ibid. P.83 (on this point he was immediately<br />
criticised by Hiebert: Hiebert "Commentary" P. 393). For the theoretical engineering see ibid.<br />
P.88: "then almost none of the pioneers used it. Instead they returned to the same older<br />
engineering tradition in which Lazare Carnot <strong>and</strong> his French successors had found the concepts<br />
needed for their new versions of the dynamical conservation theorem.". Also : "That source<br />
<strong>with</strong>in the engineering tradition is all that the pioneers of energy conservation required <strong>and</strong> as<br />
much as most of them used". Ibid. P. 84.<br />
466 Ibid. p.90. Hiebert Commentary P.394 underlines that a first concept of work<br />
meant to explain the five simple machines goes back to Hero's of Alex<strong>and</strong>ria Mechanica. But<br />
according to Cassirer the connection of the concept of work <strong>with</strong> "energy" conservation<br />
history goes back to Leibniz. Cassirer, Ernst. Leibniz' System in seinen wissenschaftlichen<br />
Grundlagen, Marburg: Elwert, 1902; Substanzbegriff Chapt 4, Sect 7, Pp.226-48; Das
assertion is surprising: it has long been known that even the third, more recent,<br />
approach had problems <strong>with</strong> the conservation of work 467. Only a fourth tradition,<br />
not mentioned in Kuhn's paper, of scientists dealing <strong>with</strong> the physicomathematical<br />
potential theory, identified the concept of work <strong>with</strong> the one of<br />
potential 468, opening the way for a mathematical expression of "energy"<br />
conservation. Thus if the concept of work is taken, as Kuhn does, in a loose<br />
sense it could derive from anybody from Hero of Alex<strong>and</strong>ria to Leibniz more than<br />
from the engineering tradition of the 18th century; in any case it would be much<br />
more a prerequisite than a trigger factor. A more stringent trigger factor or, better,<br />
a close influence is linked <strong>with</strong> a more technical concept of work, is connected<br />
<strong>with</strong> the emergence of the potential theory <strong>and</strong> is strictly related to vis viva<br />
conservation. The French theoretical engineering tradition mentioned by Kuhn<br />
has always received special attention: Ruhlmann 469 dedicated two volumes of<br />
history to the subject, Hoppe already noted 470 the influence of Carnot on<br />
Lagrange, Auerbach 471 is aware of this tradition, Cassirer 472 even discussed<br />
Poncelet's approach to projective geometry in a philosophical context. Helm<br />
Erkenntnisproblem in der Philosophie und Wissenschaft der Neueren Zeit , Berlin: B.Cassirer,<br />
vol 2.<br />
467 Helm Energetik P.12. Haas Entwickl. P.81. See also: Grattan-Guinness, Ivor.<br />
"Work for the Workers: Advances in Engineering Mechanics <strong>and</strong> Instruction in France, 1800-<br />
1830". In Annals of Science 41 (1984):1-33. P. 32.<br />
468 Wise noted the lack of mathematical factors in Kuhn's paper: Wise, Norton.<br />
"W.Thomson's Mathematical Route to <strong>Energy</strong> <strong>Conservation</strong>: a Case Study of the Role of<br />
Mathematics in Concept Formation." In HSPS 10 (1979) : 49-83. P. 5O, <strong>and</strong> attributed to<br />
Poisson the joining of the concepts of work <strong>and</strong> potential: P.64; but a different view is<br />
espressed in Wise, Norton <strong>and</strong> Smith, Crosbie. "Measurement, Work <strong>and</strong> Industry in Lord<br />
Kelvin's Britain." In HSPS 17 (1986): 147-73, P.154.<br />
469 Rühlmann Maschinenlehre<br />
470 Hoppe, E.. Histoire de la Physique . Paris: Payot,1928. P.96; compare <strong>with</strong> Kuhn<br />
Sim Disc n.44 p.86.<br />
471 Auerbach, F."Feld, Potential, Arbeit, Energie und Entropie.", in "Grundbegriffe".<br />
In H<strong>and</strong>buch der Physik. A.Winkelmann ed.2nd ed. Leipzig: Barth, 1908. 1st vol. Pp.68-91.<br />
472 Cassirer, Ernst. Das Erkenntnisproblem in der Philosophie und Wissenschaft der<br />
Neueren Zeit , Berlin: B.Cassirer. 4th vol tr by William Woglom <strong>and</strong> Charles Hendel. The<br />
Problem of Knowledge. New Haven: Yale U.P., 1950. Pp.49-50 <strong>and</strong> P.72. He discussed<br />
Poncelet's Traité des Propriétés Projectives des Figures of 1822.
underlined the acceptance of the French engineering tradition in Germany; his<br />
remarks include a review of German textbooks published before <strong>Helmholtz's</strong><br />
works 473 <strong>and</strong> these remarks are also explicitly quoted in Merz 474. As I am going to<br />
show Helmholtz was in fact aware of this more sophisticated tradition <strong>and</strong> used<br />
the term "Arbeit" ("travail") in its new technical sense in the first chapter of the<br />
Erhaltung . At variance <strong>with</strong> what Kuhn asserts, he consciously dropped it in the<br />
second chapter to introduce his "sum of tension forces". This conscious meaning<br />
shift is of great importance in underst<strong>and</strong>ing <strong>Helmholtz's</strong> paper.<br />
As to the second factor it has been pointed out that the emphasis on the<br />
interconversion of the forces of nature is not specific to the period 1830-1850 <strong>and</strong><br />
thus cannot be attributed to "the availability of the conversion processes" 475.<br />
The third factor that for Kuhn "triggered" the discovery of energy<br />
conservation is the philosophy of nature, <strong>and</strong> particularly the German movement<br />
of Naturphilosophie . But Haas' book, not unknown to Kuhn, collects a great<br />
number of 'methaphysical' contributions relevant for energy conservation, starting<br />
from Greek atomism 476. These contributions were all outlining the unity,<br />
uniformity <strong>and</strong> homogeneity of natural phenomena. Naturphilosophie too is<br />
present in the list, but <strong>with</strong>out a privileged role. Kuhn's strategy to show the<br />
relevance of his third factor is the following: this time stressing vis viva<br />
conservation 477, he (correctly) remarks that its metaphysical aspects were<br />
dropped after 1750 <strong>and</strong> returned a century later. He claims the influence of<br />
Naturphilosophie in this comeback, but he also admits that "The roots of<br />
Naturphilosophie can, of course be traced back .... to Leibniz" 478. Evidence of an<br />
influence on Helmholtz is supposedely found in a controversial remark of 1882,<br />
where Helmholtz recognizes Kantian influences in the 1847 Erhaltung 479. But to<br />
recognize Kantian roots in the Naturphilosophen is different from asserting that<br />
473 Helm Energetik Pp.14-15.<br />
474 Merz European Vol 2. P. 101.<br />
475 Heimann, Peter. "Conversion of Forces <strong>and</strong> the <strong>Conservation</strong> of <strong>Energy</strong>." In<br />
Centaurus 18 (1974):147-61. P.147 <strong>and</strong> 159.<br />
476 See the detailed analysis in Haas Entwickl ; particularly chapt 5, Pp.31-35 <strong>and</strong><br />
ch.6. Pp.35-45. Kuhn in Sim Disc quotes Haas' book in nn. 39, 74, 75, 79, 82, 92.<br />
477 "Though the technical dynamical conservation theorem has a continuous history<br />
from the early eighteenth century to the present.." Kuhn. Sim Disc p.97<br />
478 Ibid n.77 p.97<br />
479 Helmholtz WA 1 p. 68 .
Kant was a Naturphilosophe 480. The methodological role of Kantian, <strong>and</strong> even<br />
Leibnizian, influences in <strong>Helmholtz's</strong> 1847 essay ( as I am going to show:<br />
empirical <strong>and</strong> transcendental causality, conceptual explanation, regulative use of<br />
the principle of conservation) in my view cannot be confused <strong>with</strong> ontological<br />
commitments typical of Naturphilosophie.<br />
In my view Kuhn's three factors <strong>and</strong> in general his distinction between<br />
prerequisites <strong>and</strong> trigger factors are unsatisfactory <strong>and</strong> do not lead to convincing<br />
historical results. Three elements seem insufficient to explain the emergence of<br />
<strong>Helmholtz's</strong> Erhaltung <strong>and</strong> of the different formulations of "energy" conservation.<br />
Kuhn explicitly asserts 481 not to have discussed the dynamical theory of heat 482<br />
<strong>and</strong> the impossibility of perpetual motion because he considers them as<br />
prerequisites <strong>and</strong> not as trigger factors. But both were extremely relevant,<br />
specially in the case of Helmholtz, together <strong>with</strong> the central force hypothesis, a<br />
main point of debate among scientists accepting the dynamical theory.<br />
Among other short-range influences that were relevant for Helmholtz<br />
researches <strong>and</strong> are not mentioned in Kuhn's paper I want to stress the wave<br />
theory of heat 483 <strong>and</strong> the already recalled physico-mathematical tradition dealing<br />
<strong>with</strong> potential theory <strong>and</strong> the concept of work as a total differential. A detailed<br />
480 see Williams, Peirce. "Kant, Naturphilosophie <strong>and</strong> Scientific Method." In<br />
Foundations of Scientific Method: The Nineteenth Century. Ronald Giere <strong>and</strong> Richard<br />
Westfall (eds). Bloomington: Indiana U.P., 1973. Pp.3-22.<br />
481 Kuhn Sim Disc Pp.101-3<br />
482 For discussions of the role of the dynamical theory in the works of four British<br />
scientists see: Smith, Crosbie. "A New Chart For British Natural Philosophy: The<br />
Development of <strong>Energy</strong> Physics in the Nineteenth Century." In Hist Sc 16 (1978): 231-79;<br />
<strong>and</strong>: Moyer, Donald. "<strong>Energy</strong>, Dynamics, Hidden Machinery: Rankine,Thomson <strong>and</strong> Tait,<br />
Maxwell". In SHPS 8 (1977): 251-68. They differ on the priority: for Smith is Thomson's,<br />
while for Moyer is Rankine's.<br />
483 Brush showed that between 1830 <strong>and</strong> 1850 a number of physicists believed in a<br />
wave theory of heat, based on the analogies discovered by Melloni between radiant heat <strong>and</strong><br />
light. The influences on the "energy" debates are discussed, including <strong>Helmholtz's</strong> explicit<br />
references in the Erhaltung . Brush, Stephen. "The Wave Theory of Heat: A Forgotten Stage<br />
in the Transition from the Caloric Theory to Thermodynamics". In BJHS 5 (1970-71): 145-67.<br />
Reprinted in: Brush, Stephen. The Kind of Motion We Call Heat. 2vols. Amsterdam: North<br />
Holl<strong>and</strong>, 1976. Vol.2 pp. 303-25.
historical study of this tradition is still wanting 484, but its relevance for the<br />
"energy" conservation debates is great. Not only in fact in the works of<br />
Poisson 485, Green 486, Hamilton 487, Gauss 488, Jacobi, F.Neumann 489 did it precede<br />
<strong>Helmholtz's</strong> essay (who in 1847 did not know of Green but quoted Gauss), but<br />
also followed it. <strong>Helmholtz's</strong> remarks on Jacobi's appreciation of the Erhaltung<br />
for recognising its links <strong>with</strong> the older tradition are well known 490, but also I want<br />
to stress that Clausius, Weber, Kirchhoff, Riemann, C.Neumann, Sturm kept<br />
talking of "vis viva conservation" well after the "discovery" of "energy"<br />
conservation 491. To formulate a conservation principle they thought sufficient the<br />
484 But see: Auerbach, F. "Potentialtheorie". In H<strong>and</strong>buch der Physik. A.Winkelmann<br />
ed.2nd ed. Leipzig: Barth, 1908. 1st vol. Pp. 179-210. Hoppe, E. Histoire de la Physique .<br />
Paris: Payot,1928. Pp.566-74; Kline, Morris. Mathematical Thought From Ancient to<br />
Modern Times Pp.522-31 <strong>and</strong> 681-87.<br />
485 see Wise quoted above.<br />
486 Green, George: Mathematical Papers of the late George Green . N.M.Ferrers ed.<br />
London: Mac Millan, 1871."Editor's Preface" Pp.VIII-IX; "On the Laws of the Reflection <strong>and</strong><br />
Refraction" P.245, 248.<br />
P.264.<br />
487 Hamilton "On a General Method in Dynamics" 1834. Rep in Lindsay Histor<br />
488 Gauss, C.F."On General Propositions Relating to Attractive <strong>and</strong> Repulsive Forces<br />
Acting in the Inverse Ratio of the Square of the Distance". In Taylor's Scientific Memoirs<br />
Vol.II Part X. Pp.155-7.<br />
489 Neumann, Franz. Die mathematischen Gesetze der Inducirten Elektrischen<br />
Ströme C.Neumann ed. Leipzig: Engelmann, 1889. See: Olesko "Neumann".<br />
Sketch . P.12.<br />
490 Helmholtz WA1 Appendix to the Erhaltung P.74; Helmholtz Autobiographical<br />
491 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen, insbesonder uuber das<br />
Prinzip der Erhaltung der Energie." 1871. In Werke 6 vols. Berlin, 1892-4. Vol.4. Pp. 247-99.<br />
Tr. in Phil Mag 43 (1872): 1-20 <strong>and</strong> 119-49; Clausius, Rudolf. De la fonction potentielle et du<br />
potentiel. Tr. by F.Folie. Paris: Gauthier-Villars, 1870; Clausius, Rudolf. "Ueber das Verhalten<br />
des elektrodynamischen Grundgesetzes zum Prinzip von der Erhaltung der Energie und über<br />
eine noch weitere Vereinfachung des ersteren." In Pogg Ann 157 (1876):489-94; tr.in Phil<br />
Mag s5 1 (1876): 218-21; Clausius, Rudolf. Die Mechanische Wärmetheorie 2nd ed. 2nd<br />
vol. Braunschweig,1879; Sturm, Jacques. Cours de Mécanique , M.Prohuet ed. Vols. 2. Paris:<br />
Mallet-Bachelier, 1861; Riemann, Bernhard. Schwere, Elektricität und Magnetismus, nach den<br />
Vorlesungen von B.Riemann . Karl Hattendorff (ed). Hannover: Carl Rumpler,1876. P.156;
condition that work done by the forces be a total differential. This means that<br />
"energy" was not stressed as an autonomous physical concept but that what was<br />
really supposed to be important was a formal condition on the work done by the<br />
forces. An analysis of this tradition <strong>and</strong> of its relations <strong>with</strong> <strong>Helmholtz's</strong><br />
Erhaltung <strong>and</strong> later works ( <strong>Helmholtz's</strong> attempted in Königsberg at convincing<br />
F.Neumann of energy conservation 492, while was actually F.Neumann who taught<br />
the potential theory to Helmholtz 493) would outline some interesting differences<br />
between mathematical <strong>and</strong> theoretical approaches to energy conservation.<br />
Once that we recognise that something relevant happened in the history of<br />
conservation ideas in the 40's <strong>and</strong> 50's of last century more <strong>and</strong> more compelling<br />
appears that to try to clarify what happened the stress should be put on<br />
differences rather than on similarities.<br />
This leads to reexamine Kuhn's choice of the twelve "pioneers" <strong>and</strong> the<br />
three groupings. The exclusion from group one of Clausius 494, Rankine 495 <strong>and</strong><br />
Kirchhoff: see Hoppe Histoire P.99; Neumann, Carl. Die Gesetze von Ampére und Weber.<br />
Leipzig: Teubner, 1877.<br />
492 Koenigsberger HvH P.64<br />
493 Koenigsberger HvH . Pp.100-1. Helmholtz in his two main papers in<br />
hydrodynamics (Helmholtz, Hermann. "Ueber Integrale der hydrodynamischen Gleichungen,<br />
welche den Wirbelbewegungen entsprechen." In Journ. f. d. reine u. angew<strong>and</strong>te Mathematik<br />
55 (1858): 25-55; rep. in W A 1 pp.101- 134; "Ueber discontinuirliche<br />
Flüssigkeitsbewegungen." In Berliner Monatsberichte (1868): 215-28; rep. in W A 1 pp.146-<br />
157) <strong>and</strong> in his electrodynamics works of the seventies drew more on potential theory than on<br />
his 1847 version of "energy" conservation.<br />
494 On Clausius see: Helm Energetik Pp.70-80; Daub, Edward." Atomism <strong>and</strong><br />
Thermodynamics." In Isis 58 (1967): 293-303; Clark, Peter. "Atomism versus<br />
Thermodynamics." In Method <strong>and</strong> Appraisal in the Physical Sciences: The Critical<br />
Background to Modern Science, 1800-1905. Colin Howson ed. Cambridge: Cambridge U.P.,<br />
1976, pp.41-105; Truesdell, Clifford. The Tragicomic History of Thermodynamics. 1822-<br />
1854. New York: Springer,1980; Yagi, Eri."Clausius's Mathematical Method <strong>and</strong> the<br />
Mechanical Theory of Heat." In HSPS 15 (1984): 177-95; Wolff "Clausius".<br />
495 On Rankine see: Hutchinson, Keith."W.J.M. Rankine <strong>and</strong> the Rise of<br />
Thermodynamics." In Brit J Hist Sci 14 (1981): 1-26; see also Bevilacqua En <strong>and</strong> Electr .<br />
Pp.80-90.
W.Thomson 496, is striking. In fact each of these four scientists gave a formal<br />
expression of a "principle of conservation". They all gave substantial<br />
contributions in the period chosen <strong>and</strong> their relations <strong>with</strong> <strong>Helmholtz's</strong> approach<br />
are indeed relevant.<br />
The history of the experimental determination of the equivalent between<br />
work <strong>and</strong> heat is very sophisticated 497. Kuhn identifies as contributors to this<br />
determination the members of the first two groups, that is: Mayer, Joule, Colding,<br />
Helmholtz; Carnot Seguin, Holtzmann, Hirn. But, as I am going to show,<br />
Helmholtz has to be excluded: we are then left <strong>with</strong> seven "pioneers". Planck 498<br />
<strong>and</strong> later almost identically Helm 499, referring to the same period discussed by<br />
Kuhn, cited another seven authors who provided determinations: Clausius,<br />
Kupffer, Favre, LeRoux, Bosscha, Quintus Icilius, Matteucci. They discussed<br />
thirty-one different determinations, including four of Hirn's. Kuhn does not seem<br />
to be aware of these analysis: about Hirn he asserts that "none of the st<strong>and</strong>ard<br />
496 On W.Thomson see: Smith, Crosbie. "Natural Philosophy <strong>and</strong> Thermodynamics:<br />
William Thomson <strong>and</strong> 'The Dynamical Theory of Heat'." In Brit J Hist Sci 9 (1976): 293-319;<br />
Smith, Crosbie. "William Thomson <strong>and</strong> the Creation of Thermodynamics, 1840-1855." In Arch<br />
Hist Exact Sci 16 (1976-7): 231-88; Smith, Crosbie. "A New Chart For British Natural<br />
Philosophy: The Development of <strong>Energy</strong> Physics in the Nineteenth Century." In Hist Sc 16<br />
(1978): 231-79; Wise "Math Route" ; Wise, Norton <strong>and</strong> Smith, Crosbie. "Measurement";<br />
Wise, Norton <strong>and</strong> Smith, Crosbie. <strong>Energy</strong> <strong>and</strong> Empire: William Thomson, Lord Kelvin, 1824-<br />
1907. Cambridge: Cambridge University Press, 1989.<br />
497 A careful discussion of the different determinations of the equivalents was already<br />
made in1869 by Sacchetti, in 1880 by Rowl<strong>and</strong>, <strong>and</strong> also in 1906 by Graetz. See: Sacchetti,<br />
Giovanni "Considerazioni intorno all'origine della teoria meccanica del calore." Memorie<br />
dell'Accademia di Bologna VII, 2 (1869): p.149; Rowl<strong>and</strong>, Henry. "On the Mechanical<br />
Equivalent of Heat <strong>with</strong> Subsidiary Researches on the Variation of the Mercurial from the Air<br />
Thermometer <strong>and</strong> on Variation of the Specific Heat of Water." In Proc.Amer Acad (2), VII,<br />
(1880): p.75; Rowl<strong>and</strong>, Henry. Relazione Critica sulle varie determinazioni dell'equivalente<br />
meccanico della caloria . Venezia: G.Antonelli, 1882; Graetz, L. "Das Mechanische<br />
Wärmeäquivalent." In H<strong>and</strong>buch der Physik. A.Winkelmann ed. 2nd ed. Leipzig: Barth, 1906.<br />
3rd vol., 537-561.<br />
498 Planck Princip chapt. 1 exp.Pp.74-6 <strong>and</strong> 80-3.<br />
499 Helm Energetik Part1 chapt 7. exp P.34.
histories cites these (Hirn's) articles <strong>and</strong> even recognizes the existence of Hirn's<br />
claim" 500.<br />
In my view, for a correct grouping of the "pioneers", the acceptance of a<br />
principle of conversion <strong>with</strong> constant coefficients or of a principle of<br />
conservation of "energy" must be kept separate from the specific model adopted<br />
for "energy", from the experimental determination of the work equivalent <strong>and</strong>,<br />
finally, from the mathematical representation of the quantity conserved.<br />
Helmholtz in fact thought possible to formulate the principle of conservation in<br />
the Erhaltung <strong>with</strong> either heat models (caloric <strong>and</strong> mechanical), <strong>and</strong> the choice<br />
in favour of the mechanical one was made on experimental grounds. I want also<br />
to underline that the Erhaltung was largely independent from the experimental<br />
determination of the work-heat equivalent: as I am going to show, Helmholtz did<br />
not give much importance to Joule's (wrongly-translated) values. Finally,<br />
mechanical theory is not the only route to the work-heat equivalence: Mayer<br />
denied the first <strong>and</strong> determined the second.<br />
The need for these distinctions is shown in Carnot's case. Kuhn asserts<br />
that Sadi Carnot's Réflexions "are incompatible <strong>with</strong> energy conservation" <strong>and</strong><br />
"Carnot's version of the conservation hypothesis is scattered through a notebook<br />
written between the publication of his memoir in 1824 <strong>and</strong> his death in 1832." 501.<br />
Thus he considers Carnot as a pioneer of energy conservation for his<br />
posthumously published acceptance of the mechanical model of heat <strong>and</strong> of the<br />
relative determination of the work-heat equivalent (370Kgm) 502. But this result,<br />
while obviously placing Carnot among the pioneers of the mechanical theory of<br />
heat <strong>and</strong> of the determination of the work-heat equivalence, was not needed in<br />
order to consider him among the followers of "energy conservation". Given that<br />
Carnot correctly based his theory on the impossibility of perpetual motion, even<br />
his published results do not contradict the principle of "energy" conservation.<br />
They contradict the mechanical model of heat: Carnot's assumed an interpretation<br />
of "energy" (heat by temperature variation, <strong>and</strong> this "energy" equals work) that<br />
later was ab<strong>and</strong>oned because the conceptual model of heat adopted (heat as a<br />
substance) was shown to be experimentally wrong. The soundness of Carnot's<br />
500 Kuhn Sim Disc P.68 n.2.<br />
501 Kuhn Sim Disc P.93 <strong>and</strong> 67 respectively.<br />
502 At the same time Kuhn asserts that, strictly speaking, Holtzmann, who also used<br />
the caloric theory, should not be included in the list. However Kuhn includes him because he<br />
made a determination similar to Mayer's one. Ibid p.67, n.2.
published approach for "energy" conservation is shown by the fact that<br />
Clapeyron's numerical work-equivalent, if reinterpreted in the new energy model,<br />
is not far from later accepted values 503. Thus the notebook is relevant as an<br />
indication of Carnot's shift from one model to another, but not of his acceptance<br />
of the conservation principle.<br />
In my view history would be better clarified if the grouping were done<br />
differently from how Kuhn did it, that is not <strong>with</strong> the aim of asserting strict<br />
similarities between the members of a group, but <strong>with</strong> the aim of a more<br />
consistent comparison of their assertions on specific points. I can suggest for<br />
instance to compare the works of the "pioneers" on the basis of:<br />
a) the kind of general principle adopted: a correlation 504 of forces, that is,<br />
a conversion <strong>with</strong> constant coefficients 505; a conservation at positions (vis viva<br />
principle) 506; a conservation of an underlying unity during a process 507.<br />
503 Clapeyron's coefficient of 1.41Kgm ( work obtained in the passage of a calory<br />
from 1°C to 0°Celsius) is consistent <strong>with</strong> later results: if we want to shift to the other<br />
interpretation of energy (heat equals work) we have only to multiply it by 273.16 (absolute<br />
temperature of melting ice). We get 385.15 Kgm, a value not too different from Joule's. See:<br />
Planck Princip . Pp.13-4 <strong>and</strong> 188-9.<br />
504 Kuhn recognizes that a distinction has to be made between correlation <strong>and</strong><br />
conservation: Kuhn Sim Disc p.79 <strong>and</strong> 82.<br />
505 As in Roget, Séguin, Liebig, Colding, Mayer, Hirn, Grove, Faraday, Joule; on<br />
Liebig see: Kremer, Richard. "The Thermodynamics of Life <strong>and</strong> Experimental Physiology,<br />
1770-1880." Dissertation. Harvard University, 1984. Dissertation Abstracts International, 45<br />
(1985), 3731A. Pp.193-215; on Colding see: Dahl, Per. "Ludwig A. Colding <strong>and</strong> the<br />
<strong>Conservation</strong> of <strong>Energy</strong>." In Centaurus 8 (1963): 174-88; on Mayer see: Lindsay, Robert<br />
Bruce. Julius Robert Mayer, Prophet of <strong>Energy</strong>. Oxford <strong>and</strong> New York: Pergamon<br />
Press,1973; Heimann, Peter. "Mayer's Concept of 'Force': the 'Axis' of a New Science of<br />
Physics." In HSPS 7 (1976) : 227-96; on the analogies between heat model in Mayer <strong>and</strong> Hirn<br />
see: Planck Princip . Pp.75-6; on Grove see: Cantor, Geoffrey. "William Robert Grove, the<br />
Correlation of Forces <strong>and</strong> the <strong>Conservation</strong> of <strong>Energy</strong>." In Centaurus 19 (1976): 273-90; on<br />
Faraday see: Gooding, David. "Metaphysics versus Measurement: the Conversion <strong>and</strong><br />
<strong>Conservation</strong> of Force in Faraday's Physics." In Annals of Science 37 (1980): 1-29. For<br />
remarks on Kuhn <strong>and</strong> Heimann see n7 <strong>and</strong> 8 Pp.2-3, n15 P.4; for Grove, Faraday <strong>and</strong> Joule<br />
see: Heimann "Conversion" pp.148-9; for the differences between the last two, see: Cantor<br />
"Faraday <strong>and</strong> Joule on <strong>Energy</strong> <strong>Conservation</strong>". Paper presented at the Joule' s Centenary<br />
meeting, Manchester 1989. Forthcoming.
) the experimental determinations of the work equivalents.<br />
c) the conceptual model adopted for heat: caloric 508, basic quantity 509, or<br />
mechanical 510.<br />
d) the kind of mathematical formulation adopted: for instance energy as a<br />
sum of two basic forms, positional <strong>and</strong> kinetic 511 or as a product of an intensity<br />
<strong>and</strong> a capacity factor 512.<br />
Other interesting comparisons among the pioneers could be done on the<br />
basis of e) the different nationalities 513 <strong>and</strong> of f) the different generations of the<br />
scientists 514.<br />
506 That is: existence of a potential or of a generalised potential, for instance in<br />
Weber, Clausius, C.Neumann; see my <strong>Energy</strong> <strong>and</strong> Electr Pp.75-8; 98-108; 122-36; 153-165.<br />
507 According to Kun: Colding, Helmholtz, Liebig, Mayer, Mohr, Sèguin; see: Kuhn<br />
Sim Disc P.94; but at P.68 Grove <strong>and</strong> Faraday shared the same idea; Grove is supposed to<br />
announce a universal conservation principle p.70, but compare <strong>with</strong> Cantor's "Grove" p.286;<br />
also Kuhn's inclusion of Sèguin in this group is not <strong>with</strong>out contradictions: in Kuhn Sim Disc<br />
P.70 Séguin is supposed to have discussed only a special case of energy conservation <strong>and</strong> at<br />
P.76 <strong>and</strong> 81 to have ignored the new conversion processes entirely.<br />
508 as in Carnot, Clapeyron, Hess, Holtzmann; the literature on Carnot is huge; see up<br />
to 1984: Home, Roderick. The History of Classical Physics. A Selected, Annotated<br />
Bibliography. New York: Garl<strong>and</strong>, 1984. Pp.238-48; on Clapeyron, Hess <strong>and</strong> Holtzmann see<br />
Helm Energetik Pp.32-3 <strong>and</strong> 58-63.<br />
509 As in Mayer<br />
510 As in Mohr, Joule, Helmholtz, Rankine, Clausius, W.Thomson; on Mohr see:<br />
Helm Energetik part 1 chapt 3 P.9; on Joule see: Cardwell, Donald S.L.. "James Prescott<br />
Joule <strong>and</strong> the Idea of <strong>Energy</strong>." In Phys.Educ. 24 (1989): 123-7; Cardwell, Donald S.L. James<br />
Joule. A biography. Manchester: Manchester U.P., 1989.<br />
511 For instance in: Helmholtz, Rankine, Thomson.<br />
512 For instance in Rankine.<br />
513 Stress could be given for instance to the differences between British, French <strong>and</strong><br />
German institutional <strong>and</strong> philosophical backgrounds, as suggested by Elkana in: Elkana A case<br />
of? p.56; Elkana, Yehuda. The Discovery of the <strong>Conservation</strong> of <strong>Energy</strong> . London:<br />
Hutchinson Educational, 1974.<br />
514 To "highlight differences in background, education, style" as hinted by Cantor<br />
after the example of Caneva. See: Cantor Locating p.9; Caneva, K."From Galvanism to<br />
Electrodynamics: The Transformation of German Physics <strong>and</strong> its Social Context." HSPS 9<br />
(1978) 63-159.
The widespread lack of discussion in the contemporary secondary<br />
literature of the great works on the history <strong>and</strong> meaning of the principle of<br />
conservation of energy written at the turn of the century seems an (intentionally)<br />
missed opportunity 515. Detailed analysis of the works of all the "pioneers"<br />
mentioned so far appear in these "classic" works, often the clarification of the<br />
problems at issue is very sharp <strong>and</strong> the different meanings of principles, models,<br />
mathematical tools, experimental techniques are precisely outlined. This lack of<br />
discussion entails also the problem of the originality of the modern<br />
interpretations. In fact, always taking Kuhn's paper as an "exemplar", it is<br />
possible to trace back some main themes of his analysis: the refusal of the<br />
priority approach to Merz 516 <strong>and</strong> Haas 517, the relevance of the conversion<br />
processes, of the concern <strong>with</strong> engines <strong>and</strong> the related importance of the "work"<br />
concept to Helm 518, Merz 519 <strong>and</strong> Haas 520; <strong>and</strong> the "influence" of Naturphilosophie<br />
again to Haas 521.<br />
Given that the mentioned 'classic' works deal <strong>with</strong> more 'actors' <strong>and</strong> in<br />
greater detail than what Kuhn does, I am inclined to think that in Kuhn's paper<br />
the selection presented <strong>and</strong> the way in which it is presented depends on the<br />
particular problem chosen: to explain the origins of the "simultaneous discovery".<br />
An alternative starting point is then needed. How is <strong>Helmholtz's</strong> paper<br />
framed by other historians? The problem of Kantian influences has been taken up<br />
by Heimann, <strong>with</strong> an analysis centred on <strong>Helmholtz's</strong> justification of the central<br />
515 Kuhn in fact quotes Helm, Haas <strong>and</strong> Mertz, <strong>and</strong> Planck's 1887 treatise is discussed<br />
by Hiebert in: Hiebert, Erwin. The Conception of Thermodynamics in the Scientific Thought<br />
of Mach <strong>and</strong> Planck. Freiburg i. Br.: Ernst Mach Institut der Fraunhofer Gesellschaft zur<br />
Forderung der angew<strong>and</strong>ten Forschung, 1968. (E.Mach Inst. Bericht Nr. 5/68). Elkana avoids<br />
discussing Planck's treatise simply asserting that: "his (Planck's 1887) insights <strong>and</strong> explanations<br />
are far beyond any methodological school." see: Elkana A case of? P.31.<br />
516 Merz, John. A History of European Thought In The Nineteenth Century. 2<br />
vols.Edinburgh <strong>and</strong> London: Blackwood, 1903. Vol2 p.98.<br />
517 Haas Entwickl . P.98.<br />
518 Helm Energetik Pp.7-34.<br />
519 Merz European vol 2 Pp.100-1 <strong>and</strong> 104-11.<br />
520 Haas Entwickl.. Pp. 63-92.<br />
521 Ibid Pp.35-45..
force principle 522. Elkana tried to show 523 that Helmholtz was the "true"<br />
discoverer of energy conservation, but centres his attempt on the 1847 paper <strong>and</strong><br />
fails to see, among other problems 524, all the later reformulations of the principle.<br />
Cantor points at the difficulties of framing Helmholtz 525, but I think that Winters<br />
offers a good contribution in analysing <strong>Helmholtz's</strong> Erhaltung in comparison<br />
<strong>with</strong> his later works on energy 526 . Necessary I believe is also to discuss the use<br />
<strong>and</strong> modifications of the principle in the debates of the following decades 527.<br />
___________________________________<br />
From such an enlarged point of view, the whole problem of "simultaneity" <strong>and</strong> of<br />
"discovery" of "energy conservation" should be ab<strong>and</strong>oned. What Helmholtz<br />
really did in his Erhaltung was to lay the foundations of theoretical physics,<br />
through the conscious interplay of conceptual models, regulative principles,<br />
mathematical techniques <strong>and</strong> experimental results. The whole interplay was<br />
based on an explicit methodology <strong>and</strong> was meant to have the greatest generality<br />
of possible applications. From then on in physics the theory-experiment interplay<br />
was joined <strong>and</strong> often overcome by the theory-principle one. From the point of<br />
view of <strong>Helmholtz's</strong> approach, the other contributions can be seen as less<br />
sophisticated <strong>and</strong> less conscious, but not less interesting, approaches. At the end<br />
of the century the great debates of the "now mighty theoretical physics" 528 show<br />
the progress achieved: the mechanic, energetic, thermodynamic <strong>and</strong><br />
electromagnetic views of nature are the frameworks <strong>with</strong>in which basic studies<br />
on the history of energy conservation were written. I believe that contemporary<br />
history of energy studies can be enhanced if the problem is seen in terms of the<br />
522 Heimann, Peter. "Helmholtz <strong>and</strong> Kant: The Metaphysical Foundations of 'Über<br />
die Erhaltung der Kraft' ." In SHPS 5 (1974) : 205-38; P.208 n.13, <strong>and</strong> P.223.<br />
523 Elkana, Yehuda. "The <strong>Conservation</strong> of <strong>Energy</strong>: A case of Simultaneous<br />
Discovery?" In Arch Int Hist Sci 24 (1970): 31-60.; "<strong>Helmholtz's</strong> 'Kraft': An Illustration of<br />
Concepts in Flux". In HSPS 2 (1970): 263-98; The Discovery<br />
524 see: Clark, Peter. "Elkana on Helmholtz <strong>and</strong> the <strong>Conservation</strong> of <strong>Energy</strong>." In<br />
BJPS 27 (1976): 165-176.<br />
525 Cantor Locating<br />
526 Winters, Stephen. "Hermann von <strong>Helmholtz's</strong> Discovery of Force <strong>Conservation</strong>."<br />
Dissertation. The John Hopkins University, 1985.<br />
527 see my <strong>Energy</strong> <strong>and</strong> Electr Pp.59-74 <strong>and</strong> 110-21.<br />
528 C.Jungnickel,R.McCormmach, Intellectual Mastery of Nature , 2 vols, Chicago:<br />
Chicago U.P., 1986. 2nd vol.
irth (in 1847) <strong>and</strong> the development of theoretical physics, as a discipline<br />
different from the experimental <strong>and</strong> the mathematical.<br />
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aus dem Jahre 1855". In Fort d Ph 11 (1858): 361-73.<br />
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den Wirbelbewegungen entsprechen." In Journ. f. d. reine u. angew<strong>and</strong>te Mathematik 55<br />
(1858): 25-55; rep. in W A 1 pp.101- .<br />
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Helmholtz, Hermann. "Report on Sir William Thomson's Mathematical <strong>and</strong> Physical<br />
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Helmholtz, Hermann. "Über die physikalische Bedeutung des Princips der<br />
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Helmholtz,Hermann. Vorwort zu: Heinrich Hertz, Prinzipien der Mechanik.<br />
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Mechanics, Presented in a New Form. Tr. by D.E.Jones <strong>and</strong> T.Walley. London:<br />
1899. Repr. New York: Dover,1956.
Helmholtz, Hermann. Wissenschaftliche Abh<strong>and</strong>lungen. 3rd vol. Leipzig: Barth,<br />
1895.<br />
Helmholtz, Hermann. Vorträge und Reden 2nd ed. Braunschweig: Vieweg,1896.<br />
Helmholtz, Hermann. Einleitung zu den Vorlesungen über Theoretische Physik .<br />
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