Conservation and Innovation : Helmholtz's Struggle with Energy ...

Fabio Bevilacqua
Conservation and Innovation : Helmholtz's Struggle with
Energy Problems (1845-1894) and the Birth of Theoretical
Physics
Introduction .......................................................................................... 1
From Physiology to Physics (1845-46).................................................. 11
The Erhaltung and its two different conceptual roots: the
impossibility of perpetual motion and the Newtonian forces
hypothesis.(1847) ................................................................................. 22
1 The "Einleitung": intelligibility of nature and conceptual
explanation ................................................................................. 24
2 The two roots of the vis viva principle and their supposed
equivalence................................................................................. 30
3 The duck, the rabbit and the principle of energy
conservation................................................................................ 38
4 An easy start: Mechanics ......................................................... 46
5 Force equivalent of heat: a theoretical approach ....................... 47
6 Electricity, galvanism and thermo-electric currents:
Helmholtz and the batteries......................................................... 56
7 Which force equivalents for magnetism and
electromagnetism?....................................................................... 62
8 Conclusion............................................................................... 66
Questioning one root: the central force controversy with Clausius
(1852-54).............................................................................................. 68
Popular conservation of "force" : where have the central forces
gone ? (1854-64)................................................................................... 83
The first history of the conservation principle: "evolution and
development, not discovery" (1862-65)................................................. 100
A new tool : potential theory (1852-72) ................................................ 103
An energy battlefield for the electrodynamic debate (1870-75).............. 104
Unification, again ( 1884-94) ................................................................ 114
What had been achieved: a turn of the century viewpoint ...................... 117
Contemporary Historiography............................................................... 126
Bibliography ......................................................................................... 144

Introduction 1
Attempts at finding something in nature that is conserved through changes
are very old. At the beginnings of the Greek speculations they were connected
with discussions on the being and becoming, later with the ones about the
impossibility of perpetual motion, in more recent times with the role of the
principle of causality, the conservation of vis viva and of momentum, the
definition of the concept of work. During the 1840s and early 1850s numerous
formulations appeared of the "conversion" among natural phenomena and the
"conservation" of something underlying them. By the 1860, despite a certain
persistence of the term "force" ("Kraft") among a few German physicists, the
term energy was generally adopted, although it did not assume an unequivocal
meaning. 2. Histories rapidly written by the main actors 3, controversies about the
1 Financial support for this research has been provided by the Consiglio
Nazionale delle Ricerche. Earlier version of this essay were presented in Urbino
(1989), Munich (1990), Chicago (1990). I wish to thank David Cahan, Rod
Home, Stefan Wolff, Ivor Grattan Guinness, Enrico Giannetto and my wife
Leitha for their comments on earlier versions of the manuscript. A shorter version
of this essay, centered on the 1847 Erhaltung, is forthcoming in D.Cahan (ed):
Helmholtz Scientist and Philosopher, California U.P.
2Helmholtz recognised the equivalence between his own principle of conservation of
"force" (1847) and Rankine's conservation of "energy" (1853) as early as 1856 in the "Bericht
aus dem Jahre 1853". In Fort d Ph 9 (1856): 404-32, even if in relation to the idea of
correlation he kept using the term "Kraft".
3 See for instance the series of Helmholtz's "Bericht" on the theory of heat:
Helmholtz, Hermann. "Bericht über “die Theorie der Wärme“ betreffende Arbeiten aus dem
Jahre 1850-1851". In Fort d Ph im Jahre 1850-1851 7 (1855): 561-98; "Bericht aus dem
Jahre 1852". In Fort d Ph im Jahre 1852 8 (1855): 369-87; "Bericht aus dem Jahre 1853". In
Fort d Ph 9 (1856): 404-32; "Bericht aus dem Jahre 1854". In Fort d Ph 10 (1857): 361-98;
"Bericht aus dem Jahre 1855". In Fort d Ph 11 (1858): 361-73; "Bericht aus dem Jahre
1856". In Fort d Ph 12 (1859): 343-59.

different formulations 4 and priority debates 5 immediately followed. The term
"energy" did not have, in fact, a univocal interpretation. Other debates,
concentrated in the last two decades of the century, showed that these ideas had
grown, spread to all branches of physics and raised competing research
programs 6. During this period first-rank physicists wrote on the history and
meaning of the conservation ideas books that will here be defined as "classics" 7.
4 Like the one between Clausius and Helmholtz in the Annalen (1852-54): Clausius,
Rudolf. "On the Mechanical Equivalent of an Electric Discharge, and the Heating of the
Conducting Wire which accompanies it." In Tyndall and Francis Scientific Memoirs on
Natural Philosophy 1 (1853): 1-32 and 200-9; Helmholtz, Hermann. "Ueber einige Gesetze
der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-
elektrischen Versuche." In Pogg Ann 89 (1853): 211-33, 352-77. Repr. in WA 1 pp.475-519;
Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber die Erhaltung der
Kraft." In Pogg Ann 89 (1853): 568-579; Helmholtz, Hermann. "Erwiderung auf die
Bemerkungen von Hrn. Clausius". In Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882,
pp.76-93; Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber die
Erhaltung der Kraft", zweite Notiz." In Pogg Ann 91 (1854): 601-604.
5 For instance the "English (actually: British) controversy" in the Philosophical
Magazine (1862-65). See: Helm, Georg. Die Energetik nach ihrer geschichtlichen
Entwickelung . Leipzig: Veit, 1898. Pp.126-130; Lloyd, J.T. "Background to the Joule-Mayer
Controversy." In Notes and Records of the R.S. 25 (1970): 211-25.
6 See the interesting outlines in: C.Jungnickel,R.McCormmach, Intellectual Mastery
of Nature , 2 vols, Chicago: Chicago U.P., 1986, vol 2, pp.211-53; Hiebert, Erwin. "The
Energetics Controversy and the New Thermodynamics." In Duane H.D. Roller (ed).
Perspectives in the History of Science and Technology . Norman: Oklahoma U.P., 1971.
Pp.67-86.
7 See expecially: Planck, Max. Das Prinzip der Erhaltung der Energie . Leipzig:
Teubner, 1887. 2nd ed 1908; Helm, Georg. Die Lehre von der Energie, historisch-kritisch
entwickelt . Leipzig, 1887; Helm, Georg. Die Energetik nach ihrer geschichtlichen
Entwickelung . Leipzig: Veit, 1898; Haas, Arthur Erich. Die Entwicklungsgeschichte des
Satzes von der Erhaltung der Kraft , Vienna: Hölder, 1909; but also: Mach, Ernst. Die
Geschichte und die Würzel des Satzes von der Erhaltung der Arbeit . Prag, 1872. 2nd edition
Leipzig,1909. Trans Philip E.B.Jourdain. History and Root of the Principle of the
Conservation of Energy . Chicago: Open Court, 1911; Mach, Ernst. Die Mechanik in ihrer
Entwickelung historisch-kritisch dargestellt . Prag, 1883. Trans. The Science of Mechanics .
La Salle: Open Court, 1942; Mach, Ernst. Die Prinzipien der Wärmelehre, historisch-kritisch

The depth with which the literature was analysed is impressive. Not long
afterwards, at the beginning of this century, comes the period of the
philosophers 8, whose debates on the meaning of the principles of conservation
are an indication of their lasting importance. Finally comes the time of the
historians, who in the last thirty years have tried to give an account of this
complex but fascinating scientific struggle, even if, unfortunately, often with a
certain detachment, a lack of knowledge and acknowledgement, from the results
of previous periods.
Two more recent aspects of the historians' activity in the field seem
promising: first, an improvement in the efforts to (re)discover and print so far
unpublished sources preserved in the archives; as far as Helmholtz is concerned,
this "industry" is well alive, produces relevant documents 9 and raises new
expectations 10. Second, a historiographical problem shift: the focus on the
emergence and development of theoretical physics (in Germany) 11 is offering new
entwickelt , Trans. Principles of the Theory of Heat ,Historically and Critically Elucidated.
Norwell, Mass.: Kluwer, 1986; Maxwell, James Clerk. The Theory of Heat , London, 1870;
Maxwell, James Clerk. Matter and Motion , London, 1877; Ostwald, Wilhelm. Die Energie .
Leipzig: Barth, 1908; Poincaré, Henry. La Science et l'Hypothèse . Paris, 1902. Rep Paris:
Flammarion, 1968; Stallo,J.B. The Concepts and Theories of Modern Physics . London, 1882;
Steward, Balfour.The Conservation of Energy .London, 1874; Rühlmann, Moritz. Vorträge
Über Geschichte der technischen Mechanik und der theoretischen Maschinenlehre . Vols 2.
Leipzig, 1881-1885.
8 For instance: Meyerson, Emile. Identité et Realité , Paris, 1908; Cassirer, Ernst.
Substanzbegriff und Funktionsbegriff, Berlin: B.Cassirer, 1910; 2nd ed 1923.
9 Helmholtz, Hermann.Über die Erhaltung der Kraft . Christa Kirsten ed. Weinheim:
Physik-Verlag, 1983 (a transcription of the penultimate manuscript); Kirsten, Christa, et al.,
eds. Dokumente einer Freundschaft: Briefwechsel zwischen Hermann von Helmholtz und
Emil du Bois-Reymond 1846-94. Berlin: Akademie-Verlag, 1986; Cahan, David. An Institute
for an Empire. The Physikalisch-Technische Reichsanstalt. 1871-1918 . Cambridge:
Cambridge U.P., 1989; Kremer, Richard. Letters of Hermann von Helmholtz to his Wife,
1847-1859. Stuttgart: Steiner,1990.
10 David Cahan. Letters of Hermann von Helmholtz to his Father . Forthcoming
11 McCormmach, Russell. "Editor's Foreward" In HSPS 3 (1971) ix-xxiv; Olesko,
Kathrin. "The Emergence of Theoretical Physics in Germany: Franz Neumann and the
Königsberg School of Physics, 1830-1890." Dissertation. Cornell University, 1980;

and interesting results, despite the fact that the definition of "theoretical physics"
is not without difficulties. In my view the results achieved would be enhanced if a
better definition could be reached 12, outlining in a precise way that theoretical
physics is a discipline different both from experimental and from mathematical
physics 13.
In the light of these recent trends I propose that Helmholtz's Erhaltung 14
be seen not in the light of a "discovery","simultaneous" or not, of a once and for
all defined concept of "energy" or "principle of energy conservation", but as a
central step in the process of the emergence of theoretical physics, for the explicit
and sophisticated methodology outlined and the (neither experimental nor
mathematical) results achieved. Helmholtz's theoretical effort resulted in a
formulation of the principle of energy conservation based on the impossibility of
perpetual motion and on the Newtonian model of forces depending only on
positions. This allowed him to define two main forms of energy sharply split:
potential and kinetic. At the same time Helmholtz's methodology made clear the
distinction between theoretical and experimental physics and assessed a hierarchy
of interacting levels in the structure of physical theories. This methodology is also
a useful tool to analyse the different research programs that, based on the
different approaches to energy problems of the middle of last century, grew in the
following decades. In Helmholtz's long scientific activity the methodology played
a more stable role than his specific 1847 approach to the principle of energy
C.Jungnickel, R.McCormmach, Intellectual Mastery ; Wolff,Stefan."Clausius' way to the
kinetic theory-the beginnings of theoretical physics in Germany." Forthcoming.
12 C.Jungnickel and R.McCormmach's title is a quotation from Helmholtz's
autobiography, but a precise definition of the term "theoretical physics" widely used in the
subtitles ("Theoretical Physics from Ohm to Einstein; vol 1: The Torch of Mathematics 1800-
70; vol 2: "The Now Mighty Theoretical Physics 1870-1925") is lacking. References are made
to Boltzmann (1895): "Even the formulation of this concept is not entirely without difficulty,"
Vol.1 p.XV; to Wien (1915): vol.2 p.XV. Wolff too, facing the same problem, relies on
Boltzmann: n. 89.
13 To my knowledge this problem is only, and briefly, pointed to in Kuhn,
Thomas."Mathematical versus Experimental Traditions in the Development of Physical
Science." In The Journal of Interdisciplinary History 7 (1976):1-31. Rep. in Kuhn Ess
Tension ; see n.32. Pp. 64-5.
14 Helmholtz, Hermann. Über die Erhaltung der Kraft, eine physikalische
Abhandlung. Berlin: G.Reimer, 1847.

conservation. In fact the latter went through some serious modifications, also in
relation to the debates with alternative formulations, ending with a refusal of this
principle as the main research tool in physics. These very relevant conceptual
shifts of Helmholtz on the energy problems should be seen as chess moves on the
basis of a well defined methodology, lasting till the Einleitung to the Theoretical
Lectures 15 of 1894.
New trends in energy studies cannot avoid discussing Thomas Kuhn's paper on
the "simultaneous discovery" of "energy conservation", published in 1959. Kuhn's
paper is still unanimously defined as challenging, for the lack in contemporary
historiography of a rival synthesis. Nevertheless analysing Helmholtz's Erhaltung
, and the related primary and secundary literature, I started doubting of the
correctness of some of Kuhn's main historical and historiographical claims.
I will show in the last chapter that some major aspects of Helmholtz's Erhaltung
escaped Kuhn's (and other historians') attention: 1) Helmholtz's methodological
four level structure; 2) his demarcation between theoretical and experimental
physics; 3) his overcoming of both the engineering and the mathematical
approach to the work concept; 4) the lack of an experimental determination of a
work equivalent of heat and the mistranslation of James Joule's values; 5) the
difficult (and sometimes wrong) theory-experiment interplay in the application of
the principle; 6) the formulation of a lasting methodology and of a non lasting
conceptual model of energy.
My work is meant as a contribution to: a) discussing the actual content of
the Erhaltung ; b) pointing out the developments and modifications of the
subsequent works of Helmholtz mainly in relation to the debates with Rudolf
Clausius and Wilhelm Weber; c) raising the problem of a "return to the classics";
d) discussing the recent historiographical debate.
The theme of "energy" was a central one in Helmholtz's long scientific
activity. When he decided to publish his own collected papers he gave "Zu Lehre
von der Energie" the first place in the first volume 16. The three papers collected
there belong to three different stages of Helmholtz's early researches on energy:
the first, that is also the first of Helmholtz's famous Bericht , is the border-line
between the early physiological researches and the physics of the Erhaltung . A
15 Helmholtz, Hermann. Einleitung zu den Vorlesungen über Theoretische Physik .
Arthur König and Carl Runge eds. Leipzig: Barth, 1903.
1882.
16 Helmholtz, Hermann. Wissenschaftliche Abhandlungen. 1st vol. Leipzig: Barth,

principle of correlation is expressed and various applications given, but the
"energy" terms of the equations are expressed in units of heat and not work.
The Erhaltung is the full expression of Helmholtz's ideas on theoretical
physics and on energy. Its "Einleitung" is the clear and conscious indication of
the methodological control Helmholtz had achieved on his own researches.
Helmholtz outlined a four levels structure. Two basic physical hypotheses
(impossibility of perpetual motion and central Newtonian forces) were placed at
the first level, "the" principle of conservation at the second, empirical laws at the
third and natural phenomena at the fourth. Moreover the two basic hypotheses,
echoing various elements of Kantian philosophy, were not presented as self
evident but as the result of a philosophical 'explanation'..
In my view the need that Helmholtz felt to justify his own version of the
conservation principle on higher grounds (on two physical hypotheses in turn
justified on philosophical grounds) is a clear indication of Helmholtz's
consciousness of the possibility of alternative formulations of the principle itself.
Helmholtz not only wanted to express a principle, but also to establish the
framework and the rules following which principles could be formulated and
used. This is what makes Helmholtz's approach mark a major step in the
emergence of theoretical physics, and shows that his version of the principle was
not only the application of a (meta)physical assumption, but the application of a
sophisticated methodology. The two physical assumptions (first level) are meant
to bring together, not without problems, two different but well known traditions
in physics 17, and thus to offer secure grounding for the whole enterprise.
Helmholtz believed that this was not enough and decided to justify the first level
on more abstract grounds: he connected the principle of impossibility of perpetual
motion with the principle of sufficient reason, a transcendental condition for the
intelligibility of nature, and gave a conceptual explanation of the model of central
forces in the Kantian style. Finally he hinted at an "empirical" principle of a cause
effect relationship to be embedded in the formulation of the principle of
conservation (second level). This principle, in turn, had to be compared with
existing empirical laws (third level) and to predict new ones, to eventually
achieve the intelligibility of natural phenomena (fourth level).
17 In this context I think that Elkana's suggestion of Helmholtz unifying Newtonian
and analytic mechanics can be accepted: Elkana The Discovery ; but for a better
characterization of trends in mechanics see: Grattan-Guinness, Ivor. "The varieties of
mechanics by 1800", paper for the Hegel and Science Conference, typescript, 1989.

With reference to these four levels Helmholtz could draw an explicit
distinction between theoretical physics (dealing with deductions from level two to
three, that is with the applications of the principle to empirical laws) and
experimental physics (dealing with the inductions from level four to three, that is
from natural phenomena to empirical laws). It is in my view important to remark
that the dividing element between theoretical and experimental physics is not
meant to be the use of mathematics. Thus theoretical physics was not at all
identified with mathematical physics.
In my opinion, with this Introduction, Helmholtz marks explicitly the
emergence of theoretical physics, as a discipline distinct not only from
experimental physics but also from mathematical physics. The most striking
characteristic is Helmholtz's consciousness of the role of the four levels of the
hierarchy, a consciousness that allowed him later to modify some specific
elements of his scheme, leaving the methodological structure unchanged. The
great and successful novelty is the stress on the interplay of the second and third
level : since 1847 physical laws (level three) had to satisfy more and more not
only experiments and natural phenomena (level four), but also theoretical
principles (level two). These principles were also to be seen as heuristic tools to
"discover" empirical laws: a complete new field of inquiry is open, theoretical
physics, which is not primarily based on an extended use of mathematics. A basic
characteristic of the principles is to be general, that is, to unify the different
branches of physical knowledge.
Helmholtz's methodological worries show also that he was obviously
aware of the difficulties of his gigantic plan to unify natural sciences under one
regulative principle. As I try to show below the Erhaltung is in fact "only" a plan
that requires corroboration. If a discovery was made by Helmholtz in 1847, it
was not that (of a specific formulation) of energy conservation, still a theoretical
proposal lacking new experimental data and a secure mathematical grounding 18,
but that of theoretical physics.
In the Erhaltung an effort at showing the equivalence between the two
main hypotheses (level one) is made in the first chapter, while the deduction from
these hypotheses of Helmholtz's version of the principle of conservation (level
two) is the object of the second chapter. The following four chapters deal with
the interactions between levels two and three, that is with the application of the
18 See below the difficulties with the geometrical interpretation of the integral, in
section 3 of the Erhaltung, and with the definition of selfpotential, in section 6.

principle to the existing empirical laws and with the attempt at deriving new
empirical laws on theoretical grounds. In the Erhaltung no new experimental
data (level four) are offered and the few regarding the mechanical equivalent of
heat available at the time are criticised, also on the basis of a mistake on the
conversions of the units of measurement, and disregarded.
What Helmholtz really did in his Erhaltung was to lay the foundations of
theoretical physics, through the conscious interplay of conceptual models,
regulative principles, mathematical techniques and experimental results. His
formulation of the energy principle was a specific application of this wideranging
methodology and thus was meant to have the greatest number of possible
applications.
But Helmholtz's foundations of theoretical physics while being connected
are not coextensive with his approach to energy conservation and will survive
the latter. Helmholtz's methodology was in fact extraordinarily successful.
Energy theory, that is theoretical physics applied to energy problems,
changed the shape of physics : after 1847 physical laws no longer had only to
face the challenge of experimental results, but also had to be judged on more
theoretical grounds, in their relation with the conservation principle. But in turn
the principle had, from the beginning, different formulations. From the point of
view of Helmholtz's methodology, the other formulations of the principle of
energy conservation can be seen as less methodologically sophisticated, but not
less physically interesting. From 1847 on in physics the traditional theoryexperiment
interplay was united with and often overcome by theory-principle
interplay.
Poggendorff's refusal of the paper for publication in the Annalen is an
indication of the difficulty for experimental physicists to accept the new ideas,
but it did not last long. An English version of the Erhaltung appeared in 1853
and the third paper of the energy section of Helmholtz's collected works
mentioned above reveals that, just a few years after Poggendorff's refusal, the
controversy with Clausius (1852-54), which revealed the first deep differences
between a theoretical and a more mathematical approach, was entirely published
in the Annalen . Moreover in six reports (1855-59), written for the Fortschritte
der Physik and dealing with the "Theorie der Wärme", Helmholtz discussed and
compared his own approach with an already wide literature: this might be
considered the first history of energy conservation. Between 1862 and 1865, in
the Philosophical Magazine , an international controversy on the "priority" of the
"discovery" showed that the topic had become basic in scientific research.

Helmholtz himself, starting in 1854, was playing a major role in
popularising the new view, but, surprisingly, almost hiding his initial conceptual
model of central Newtonian forces and stressing instead the other assumption, the
impossibility of perpetual motion. This shift, which again demonstrates
Helmholtz's control of the various interacting elements, is in my view the result of
Clausius' criticisms, but also of a growing interest in potential theory and in its
applications, for instance in hydrodynamics (1858 and 1868) 19. Also relevant are
the difficulties created for Newtonian forces by the growing field of
electromagnetic phenomen. The debates of the seventies underline the
methodological relevance of the 'energy principle', now a main tool for
comparison of experimentally equivalent alternative theories, but at the same
time outline the problem of identifying 'the' principle: various versions were, as
usual, competing. Helmholtz's version reveals some limits of the original 1847
assumptions.
In these years Helmholtz faced the challenge of the nativists and the
metaphysicians and, also as a result of his work on physiology and the
foundations of geometry, shifted towards a more empirical side of his
methodological approach on many issues. A strict adherence to Kantian
categories and Kantian conceptual models was abandoned, for instance the
central Newtonian forces, while the methodological Kantian structure of the
Erhaltung is preserved, in particular the need for a regulative principle that
fulfils the "intelligibility of nature".
In the last two decades of the nineteenth century the history of energy
conservation entered a new phase: the differences between the original versions
of the principle grew into different research programs. The mechanical view of
nature and the belief in the reversibility of natural phenomena championed by
Helmholtz were challenged by other, competing views. The energetist movement
took off, with its attempts to overthrow the mechanical worldview, while the
electromagnetic and the thermodynamic approach, both deeply connected with
energy problems, were actively pursued. Since energy conservation had become
one of the basic chapters of physics, the champions of the different research
programs dedicated a great deal of logical and historical analysis to an
understanding and framing of the various contributions and developments of
energy theory. Given that a specific version of the conservation principle played a
great role in each research program, a series of works started to analyse in detail
19 Helmholtz "Integrale" and " Flüssigkeitsbewegungen".

the origin and meaning of the principle, from different perspectives, of course.
Helmholtz was a leader in the spread of theoretical physics in the second half of
the century, and it is interesting to find out what was the judgement of his fellow
scientists on his approach to the principle of energy conservation. At the turn of
the century the great debates of the "now mighty theoretical physics" showed the
progress achieved in this field: the mechanic, energetic, thermodynamic and
electromagnetic views of nature were the frameworks within which basic studies
on the history of energy conservation were written.
But already in the eighties new interpretations of electromagnetic energy
had opened the way to Helmholtz's shift towards Maxwell's theory of contiguous
action and to the abandonment of the traditional continental direct action at a
distance approach (even if Helmholtz always translated Maxwell in his own
terms of action at a distance plus dielectric). At the same time Helmholtz
renounced the principle of energy conservation as the main basic tool in physical
research and substituted it with the principle of least action (1886) 20. But after
almost fifty years since the Erhaltung the basic aspects of his methodology were
explicitly reasserted in a discussion of the foundations of theoretical physics 21.
Helmholtz's latest results will have an influence almost exclusively on Hertz, but
his continuous struggle shows a great methodological consistency. The
"Einleitung" to the Lectures on Theoretical Physics of 1894 strongly resembles
the methodology expressed in 1847.
A few years later philosophers took an interest in the problems of the history and
foundations of energy conservation, and again Kantian influences were at issue.
In recent times, towards the end of the neo-positivist trend historians again
started to offer contributions on this important topic: perhaps it is time to try to
put these latest efforts in historical perspective.
From Physiology to Physics (1845-46)
20 Helmholtz, Hermann. "Über die physikalische Bedeutung des Princips der kleinsten
Wirkung." In Crelle's Journal 100 (1886) :137-166 and 213-222; Repr. in WA 3. Pp.203-248.
21 Helmholtz Einleitung

In 1847 Helmholtz, a military surgeon in Potsdam, published two papers,
both dedicated to energy conservation 22. Helmholtz had been a student of Müller
in Berlin and was still closely in touch with some of his best pupils, namely
DuBois-Reymond, Brucke and Ludwig. The latter together with some of Magnus'
students, namely Gustav Karsten, Wilhelm Beetz, Wilhelm Heintz and Hermann
Knoblauch, in 1845 founded the Physikalische Gesellschaft. The society started
publishing the Fortschritte der Physik and Helmholtz's "Bericht" of 1847, written
at DuBois's request, was the first of his numerous contributions to that journal.
Physiology was, without doubt, the professional and research context in which
the two 1847 works on energy were written: Helmholtz's three works 23 published
before 1847 and the two 24 in 1848 were dedicated to physiological problems 25.
22 In the first volume of his Wissenschaftliche Abhandlungen of 1882 ( WA 1 )
Helmholtz gave the first place to the section : "Zuhr Lehre von der Energie". Here he included
three works: Helmholtz, Hermann. "Bericht über “die Theorie der physiologischen
Wärmeerscheinungen“ betreffende Arbeiten aus dem Jahre 1845." In Fort d Ph im Jahre 1845
1 (1847): 346-55; repr. in WA1 , 1882, Pp.3-11; Helmholtz, Erhaltung . Rep. in WA 1 , 1882,
Pp.12-75; Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, Pp.76-93.
23 Helmholtz, Hermann. "Ueber das Wesen der Fäulniss und Gährung". In
Müller's Arch (1843): 453-62; repr.in WA 2, 1883, Pp.726-34; Helmholtz,
Hermann. "Ueber den Stoffverbrauch bei der Muskelaction". In Müller's Arch
(1845): 72-83; repr. in WA2 , 1883, Pp.735-44; Helmholtz, Hermann. "Wärme,
Physiologisch". In Encyklopädisches Wörterbuch der medicinischen
Wissenschaften, herausgegeben von Professoren der medicinischen Facultät zu
Berlin. Vol.35, Berlin: Veit &Co,1846. Pp.523-67. Repr. in WA 2, 1883, Pp.680-
725.
24 Helmholtz, Hermann. "Ueber die Wärmeentwicklung bei der Muskelaction". In
Müller's Arch (1848): 144-64; repr. in WA2, 1883, Pp.745-63. It had been presented on the
12th of November 1847 to the Physikalische Gesellschaft; Helmholtz, Hermann. "Bericht über
“die Theorie der physiologischen Wärmeerscheinungen“ betreffende Arbeiten aus dem Jahre
1846". In Fort d Ph im Jahre 1846 2 (1848): 259-60.
25 The three works of 1843, 1845, 1846 and the first of 1848 were published in
physiological journals and in a medical encyclopedia and reprinted in the "Physiologie" section
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
also the root or one of the roots of Helmholtz's formulation of energy
conservation, has been recently discussed at length 26. Helmholtz's own point of
view, expressed in 1882 and 1892 27, is that his interest in energy conservation did
not arise from empirical problems in physiology but from the inclination he had
acquired from a very early age in favour of the principle of the impossibility of
perpetual motion 28. Autobiographical reconstructions of scientists are often
unreliable, but this antiempirical remark of Helmholtz was expressed at a time
when he was stressing empirical elements in science 29 and thus deserves
attention. In fact in the early eighties, after the controversies with Friederich
Zöllner and Karl Eugen Dühring, Helmholtz underlined, at variance with his own
previous judgements, the empirical components in the early formulations of the
principle of conservation.
Physiology thus, according to Helmholtz, offered only the battleground
for an explanation of animal heat based on the principle of impossibility of
perpetual motion and on the consequent refusal of vital forces.
One of the problems of mid-century German physiology was in fact the
acceptance or refusal of vital forces in the explanation of the origins of animal
heat 30. Liebig played a great role in this debate: already in 1841 he asserted a
Fortschritte der Physik, the Erhaltung was presented to the Berlin Physikalische Gesellschaft
on the 23rd of July and submitted to the Annalen der Physik.
26 R.Kremer in his perceptive dissertation: "The Thermodynamics of Life and
Experimental Physiology, 1770-1880." Harvard University, 1984, Pp.190-3 contrasts the
"standard" view that researches in energy conservation were motivated by physiological
problems.
27 Helmholtz, Hermann. "Über die Erhaltung der Kraft" in WA1 , 1882, P.74, and
Helmholtz, Hermann. Autobiographical Sketch , Pp.10-12.
28 Koenigsberger accepted this approach: Koenigsberger, Leo. Hermann von
Helmholtz. Tr. by Frances A. Welby. Oxford: Clarendon, 1906. P.8 and pp.25-6; Kremer gives
a different interpretation of Helmholtz's remarks: he is more interested in denying the
relevance of vitalism in physiological debates than in the role ot the principle of impossibility
of perpetual motion in Helmholtz's work; see:"Therm of Life" pp.237-238.
29 Helmholtz "Über die Erhaltung der Kraft" in WA1 , 1882, Pp.71-4 and "Robert
Mayer's Priorität". In Vorträge und Reden. 2 Vols. Braunschweig: Vieweg, 1884.
30 On the relevance of the problem see: Lenoir, Timothy. The Strategy of Life:
Teleology and Mechanics in Nineteenth Century German Biology . Dordrecht and Boston:

principle of correlation of forces, that is of conversion with constant coefficients:
" From nothing no force can be generated..." 31 and in 1842, in his Tierchemie ,
refused the idea that vital forces could generate animal heat 32.
Helmholtz was greatly influenced in his first physiological researches by
Liebig's approach 33, even if he eventually rejected it 34. Helmholtz's first two
works had controversial results 35, nevertheless in 1845, following Liebig, he was
explicit on the possibility of a common origin of mechanical forces and heat
produced by an organism 36 and wondered whether this origin could be entirely
attributed to metabolism, thus avoiding vital forces.
Reidel, 1982. Pp. 195-6, 215-7, 230; his approach differs from Kremer's, see n.26. See also
Olesko & Holmes "Experiment" P.12.
31J.Liebig: Chemische Briefe, tenth letter in the 1845 edition; twelfth letter in the
third edition (Heidelberg 1851) at pp. 116-118. First printed in the supplement to Allg.Ztg. 30
Sept 1841. This is a famous passage: quoted in Helm Energetik P 10; Haas Entwickl P.57;
Kuhn Sim Disc P.95.
32 See Kremer: "Therm of Life", Pp.204-9.
33 Kremer even asserts that "Every physiological question Helmholtz considered
before 1847 had been thoroughly defined by Liebig": "Therm of Life" P.238. Liebig is often
credited with being a pioneer in the history of energy conservation. See Planck Princip P.33;
Helm Energetik P.10 ; Haas Entwickl P.57; more recently Kuhn Sim Disc P.68; Kremer
"Therm of Life" Pp.198-215.
34 Lenoir Strategy P.196.
35 The first one, on fermentation and putrefaction, was meant to support Liebig's
antivitalist position. For Koenigsberger H v H P.27 and Kremer ibid Pp. 239-40 the paper
raised confusing conclusions, while for Lenoir The Strategy P. 197 and Yamaguchi:
Yamaguchi, Chuhei. "On the Formation of Helmholtz's View of Life Process in His Studies of
Fermentation and Muscle Action - In Relation to His Discovery of the Law of Conservation of
Energy." In Historia Scientiarum 25 (1983) : 29-37,was important, stimulating and also a
good device for further research. In his second work, on metabolism during muscular activity,
Helmholtz was inconclusive as far as metabolism was concerned, also for he lacked an exact
relation between the muscular action and the heat developed. See Koenigsberger H v H P.32,
Kremer "Therm of Life" P.243, Lenoir Strategy P.202.See also Olesko & Holmes
"Experiment" P.1.
36 "whether or not the mechanical force and the heat produced in an organism could
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
the methodological strategy of the Erhaltung can already be identified.
Helmholtz accepted Liebig's principle of force correlation and his theory
on the chemical origin of heat, but asserted that two important points needed to
be clarified for a satisfactory application of the principle: the conceptual model of
heat and the definition of the heat equivalents which are utilized in the
correlation. Liebig's theoretical determinations in fact did not agree with
experimental results.
To question the conceptual model of heat as caloric might appear
counterproductive in a discussion on the origin of animal heat: Helmholtz himself
pointed out that this model was very useful to refute vital forces. In fact the
conservation of matter assured that the amount of (latent) heat ingested was the
same as that emitted by the living bodies. Even stronger support was given by
German Ivanovich Hess's law asserting that in chemical transformations the order
of the intermediate steps had no influence for the final emission of heat. Caloric
was a definite, conserved quantity. But in a strategy that wanted to take into
account the progress of all the sciences, and particularly of the physical ones, this
model had to be changed: the recent identification of the thermal radiation with
light, and the generation of heat that could not be ascribed to the liberation of
latent heat, for instance in electrical processes, compelled to accept the idea of
heat as movement 38.
The new model was not without disadvantages: it added some problems
to the refusal of vital forces. The total amount of heat was no longer considered
constant and the production of heat through the action of forces was now
admitted as possible. In principle vital forces could be considered among the
forces producing heat 39. To deny this and to reassert the principle of impossibility
of perpetual motion made compelling the solution of the second problem outlined
above: the redefinition of the heat equivalents.
attributed to the action of a vital force, specific to the organic life: "Muskelaction" WA 2
P.735; see also Koenigsberger H v H P. 31 and Kremer "Therm of Life" Pp.240-1.
37 Helmholtz:"Wärme, Physiologisch"; WA 2 Pp.695-700.
38 For the role of the undulatory theory of heat see: Brush Kind of Motion .
39 "to the physiologists that connect the essence of life with this very
incomprehensibility, we have nothing to contrappose from a theoretical point of view";
Helmholtz W A 2 P.700.

The amount of animal heat predicted by Liebig was in fact smaller then
the one measured by Piere Louis Dulong and Cesar-Mansuète Despretz. Thus the
difference might have been explained through the effect of vital forces. Helmholtz
wanted to eliminate the discrepancy between theoretical predictions (Liebig) and
experimental results (Dulong and Despretz) in order to eliminate any role for the
vital forces. He tried to achieve that through a theoretical reformulation of the
terms of both sides of the equation relating the heat ingested with the one emitted
by living bodies. Helmholtz proposed first that the heat of the "Ingesta" be
considered no more as the one resulting from the oxidation of the elements of the
food, but instead as the one resulting from the oxidation of the compound
themselves of the food. The last one was supposed to be greater. Second
Helmholtz proposed that the heat developed in the animals be considered not
only as the one produced in the respiratory organs 40, but that also the heat
produced in the blood and tissues should be taken in account 41. With this new
elements, according to Helmholtz, theoretical predictions would satisfy
experimental data and the acceptance of the new model of heat would leave no
room for the vital forces. Still the experimental corroborations were highly
problematic 42.
I want to note that the mechanical equivalent of heat was
discussed 43, even if a determination of the equivalent itself was lacking and thus
a component was lacking in the theoretical energy balance put forward, namely
the work done by the animals 44. Thus in my view Kuhn's assertion that in1845-6
"Helmholtz fails to notice that body heat may be expended in mechanical work" 45
requires qualifications: it does not imply that "Helmholtz's conservation ideas
were not complete till 1847". In fact, already in his "Muskelaction", Helmholtz
asserted that the problem was "whether or not the mechanical force and the heat
produced in an organism could result entirely from its own metabolism" 46 and in
n.148 p.248.
40 For Kremer this does not correspond to Liebig's conception: "Therm. of Life"
41 Lenoir Strat of Life P.204
42 Kremer: "Therm of Life" P.251
43 Helmholtz: WA 2 Pp.699-700.
44 Kremer asserts that Helmholtz never succeeded in bringing heat and work
operationally together into physiological research: Kremer "Therm of Life" P.238.
45 Kuhn Sim Disc p.95 n 68.
46 see above n.36.

his "Wärme" that one of the differences between the kinetic and the caloric
theory of heat is "the determination of the equivalent of the heat that can be
produced through a given quantity of a mechanical or electric force" 47. A more
detailed discussion, as shown below, is in the "Bericht". It is true that Helmholtz
did not provide in the first papers a mechanical equivalent of heat, but not for the
lack of adequate conceptualization. He lacked experimental reliable data, a
problem still present in the Erhaltung . Helmholtz himself, in the last section of
the Erhaltung , explained that given that the amount of work produced by
animals compared to the heat is small 48, it can be neglected and the problem of
conservation of force in physiology is reduced to the problem whether the
combustion and the transposition of food can produce the same amount of heat
produced by the animals. Helmholtz added that the results of his own work in the
"Wärme" and the "Bericht", compared with Dulong and Despretz's
measurements, allowed a positive answer, at least approximately.
Thus Kuhn's remarks, prompted by his desire to stress presumed
influences of Natürphilosophie on Helmholtz, seem dubious. Even contradictory
if compared with his other remarks 49, written to deny that Helmholtz had been
influenced by the concepts of conservation present in the tradition of analytical
mechanics, that Helmholtz's conservation ideas pre-existed his 1842 reading of
D.Bernoulli.
A great step forward in the elaboration of the methodological strategy was
achieved in October 1846 50 in the first "Bericht" written for the Fortschritte,
where it was published in 1847. The word "methodological" must be stressed
because despite Koenigsberger's assertion 51 that Helmholtz at the time was
involved in detailed experimentations, no new experimental result whatever is in
this paper. The relevant feature is the extension of the correlation principle from
physiology to various branches of physics and chemistry and thus this was
Helmholtz's first attempt at a general application of his methodology. Without
47 Helmholtz "Wärme" WA2 Pp.699-700.
48 See also the views expressed in the "Wärme" quoted in Olesko & Holmes
"Experiment" Pp.21-2 n.70.: "(animal heat) is by far the greatest part of the Kraftequivalent".
49 Kuhn Sim Disc p.97 n.76.
50 Koenigsberger H v H P.34.
51 ibidem pp.34-5.

doubts this "Bericht" is a relevant work, even if it has not yet received the
attention it deserves 52.
Summarising the problem discussed in his 1846 paper, Helmholtz
explicitly asserted that heat cannot originate out of nothing (nicht aus nichts ) and
stated a principle of correlation with Liebig's words: Das Princip von der
Constanz des Kraftaquivalents bei Erregung einer Naturkraft durch eine andere
53 (the principle of the constancy of the force-equivalents for the stimulation of
one force of nature through another). But Helmholtz, while retaining the principle
of correlation, criticised its application to animal heat. As already mentioned
Liebig's solution did not correspond to the calorimetric results of Dulong and of
Despretz: the direct measurement of the combustion heat of the quantities of
hydrogen and carbonium corresponding to the food ingested was only from 70 to
90% of the heat provided by the animals.
But the correlation principle should not be confined to the problem of
animal heat. It is in fact based on the impossibility of perpetual motion, that
"logically is completely justified", and has already been applied in the
"mathematical theories" of Carnot-Clapeyron (still based on the conceptual model
of the caloric) and of Franz.Neumann (based on the concept of electrodynamic
potential). Nevertheless, Helmholtz remarked, the principle has not yet been
expressed in complete form nor experimentally verified, despite a complete
corroboration from the experimental results already achieved. Helmholtz saw
correlation as a more sophisticated expression of the principle of impossibility of
perpetual motion and immediately utilised it, offering a series of energy balances
based on the new model of heat as movement. On the basis of both the
"constancy of the force-equivalents" 54 principle and of the mechanical model of
heat 55, it must result that "mechanical, chemical and electric forces can always
few lines.
52 For instance Lenoir does not deal with this paper and Kremer dedicates to it only
53 Helmholtz W A 1 P.6. See also P.4 (reference is here to Liebig's 1845 "Ueber die
tierische Wärme").
54 Helmholtz W A 1 P.6
55"But at present the material theory of heat is no more acceptable, but should be
substituted with a kinetic one, for we see that heat originates from mechanical forces either
directly, i.e. through the friction of solid bodies against solid bodies and of fluids against
solids, or indirectly through electrical currents,from the movement of magnets and from

generate a determined equivalent of heat, however complicated the transition
from one force to the other" 56. Helmholtz asserted that the empirical evidence
was not very large but still he wanted to offer specific theoretical applications of
the principle to the heat produced through mechanical, chemical, electrolytic and
electrostatic forces. The case of animal heat was now only the last of five
applications of a principle that was becoming more and more general.
A great difficulty immediately appeared: the most important balance, the
one between heat and work, could not be written for the lack of a mechanical
equivalent of heat; in fact the values offered by the theories of Carnot-Clapeyron
and Holtzmann could not be accepted, being based on the caloric model and only
referred to the propagation and not to the production of heat 57. Helmholtz, despite
having clearly focussed the problem, lacks experimental data. In October 1846 he
still did not know of Mayer's nor of Joule's work: "no experiment can be taken in
account for the mechanical forces" 58. Thus all the other balances were written as
equivalences based on heat units and not on work units. In the "Bericht" heat and
not work was the unity of measurement common to all the natural phenomena
considered, not a minor difference with the subsequent Erhaltung.
In the analysis of chemical transformations Hess' law of the constancy of
the production of heat, whichever the intermediate stages of the reaction, was
acknowledged. Here appeared the first instance of the identification of latent heat
with the thermal equivalent that was to play a role in the subsequent discussion of
animal heat.
For the electrolytic currents the heat developed in the circuit must be
equivalent to the electrochemical transformations in the galvanic chain (battery),
independently of their order. The heat in the circuit could be calculated through
Georg Simon Ohm's and Emily Christianovic Lenz's laws (Joule was not
mentioned):
frictional electricity, where a release of latent heat is inconceivable". Helmholtz: W A 1 Pp.6-
7.
56 Helmholtz W A 1 P.7
57 It is interesting to remark that instead in the Erhaltung these data were utilised,
that Clausius criticised this "improper" use, that Helmholtz accepted the criticisms, that
Truesdell denies the validity of these criticisms. See below.
58 Helmholtz WA 1 P.7.

where H is the heat, J the intensity of the electric current, W the total
resistance of the circuit; on the other side, through Faraday's electrolytic law, we
have:
H=AC
where A is the electrical "difference" 59 of the metals involved and C is the
quantity of atoms "consumed" (atoms that underwent a process of oxidation and
reduction). According to the principle of equivalence, the heat produced in the
circuit must be equivalent to the one that could be produced through the
electrochemical transformations in the cells.
For static electricity Helmholtz stated, in a couple of lines, that the
production of heat through an electric discharge follows from Pieter Riess's
principles; thus a balance was established between the resulting heat on one side
and the product of the quantity of electricity by the electrical density (a still
Voltian term for what was later identified with the tension or the difference of
potential) on the other. Helmholtz's terminology in 1846 was still influenced by
Volta's idea of tension and far from potential theory.
Finally discussing animal heat, Helmholtz identified the latent heat of
chemical reactions with the thermal equivalent that could be produced in further
reactions. The energy balance must hold between the latent heat of the "Ingesta"
on one side and the heat "provided by the animals" plus the latent heat of the
"Egesta" 60 on the other side. The great contribution of Helmholtz is that the
equivalent at the first side of the balance is no longer the "heat of combustion of
hydrogen and carbonium but that of the food" 61. Through this reformulation and
the modification of the respiratory theory, Helmholtz claimed to have achieved a
refutation of vital forces while being in agreement with Dulong's and Despretz's
experimental results, and thus to have overcome Liebig's difficulties.
It is evident that a new methodology had been acquired and that
Helmholtz was aware of its great generality. The "Bericht" is in fact the border
line between physiology and energy conservation; it is relevant to note that in
many respects the methodology of the "Bericht" is very close to the one of the
subsequent Erhaltung : to enunciate a principle, a conceptual model of the
quantities involved, to express an equation between the energy terms and then to
compare it with the empirical laws. There are some important differences : the
59 Helmholtz WA 1 P.7
60 Helmholtz WA 1 P.8
61 Planck Princip P.34.

"Bericht", despite the application of the principle to an analysis of some physicochemical
laws, is still largely dedicated to physiology. In the much longer
Erhaltung , instead, physiology is confined to a few lines at the end of the last
chapter. In the "Bericht" the equivalence principle based on the impossibility of
perpetual motion (but also on the opposite impossibility of destroying motion:
nothing can be created and nothing can be destroyed; a conservation of the
coefficients of correlation applies) is present together with a model for many
equivalents (the terms of the energy balance), but the equivalence principle, i.e.
the correlation principle applied in the "Bericht" is very different from the
mechanical principle of conservation of energy expressed in the Erhaltung. The
equivalence principle is much closer to the ideas of Mayer and Joule, despite the
fact that in 1846 Helmholtz did not know of their works (none of them was cited
in the "Bericht"). In fact it only asserts the numerical equivalence of the effects
involved and does not imply the assumption of central Newtonian forces and that
every effect must have a mechanical interpretation in terms of potential and
kinetic energy 62. A final relevant point is that in the "Bericht" Helmholtz did not
discuss the specific determinations of the mechanical equivalent of heat, despite
his acceptance of the mechanical theory.
In 1847 Helmholtz, while writing the Erhaltung, worked out a sixth
paper 63 again dedicated to physiological problems, later to be reprinted in the
"Physiologie" section of his collected papers..
Here Helmholtz finally tried to link the problem of animal heat with that
of the mechanical force produced by muscle action. Relevant to his purposes was
to demonstrate that heat is produced in the muscle itself. He devised a very
sensible thermocouple which, linked to an astatic galvanometer and a magnifying
coil, could detect differences of temperature in the range of one thousandth of a
degree centigrade. Through thorough experiments on frogs' legs Helmholtz was
able to find evidence that heat is generated directly in the muscle tissue, that its
origins are due to chemical processes and that production of heat in the nerves is
negligible (and thus vital force could be disposed of on empirical grounds). The
role of this experimental research on the sources of animal heat, carried forward
together with and immediately after the writing of the Erhaltung, is very
62Thus I cannot agree with Lenoir's assertion:"the physiology of muscle action laid
before Helmholtz all the elements of conservation of energy" Lenoir Strat of Life P.211.
63 Helmholtz "Wärmeentwicklung". A detailed analysis is given in Olesko & Holmes
"Experiment" Sect 6.

important also for Helmholtz's understanding of energy conservation: it was in
fact the only experimental research in this field that he made. Helmholtz's
understanding and evaluation of the mechanical equivalent of heat is probably to
be connected with this very research 64.
In the next section I will show that the entire Erhaltung only offered a
theoretical reinterpretation of known results, but no new experiments. In my view
physiology could not and did not provide a key to Helmholtz's conservation of
"force": it did not offer theoretical nor experimental evidence for the
establishment of the principle of force correlation (the refusal of vital force was,
rather, based on an already accepted impossibility of perpetual motion).
Nevertheless it did provide Helmholtz with an interesting battleground and his
only original experimental data 65.
The Erhaltung and its two different conceptual roots: the
impossibility of perpetual motion and the Newtonian forces
hypothesis.(1847)
From October 1846 to July 1847 Helmholtz, not distracted but rather
inspired by his love and engagement with Olga von Velten 66, worked hard at the
Erhaltung. But we already know that this was not his only commitment: at the
same time he was also involved with experiments on the heat produced during
muscular action. Koenigsberger asserts that in the first quarter of 1847 Helmholtz
had formulated his ideas on energy conservation and had tested them with
64 This point is made by Lenoir ibid. P.211, even if details are not given. This
interpretation offers a clue to Koenigsberger's already recalled claims of deep experimentation
carried forward by Helmholtz in 1846-47. See also Kremer ibid.P.244. Olesko & Holmes
"Experiment" P.34.
65 See also Koenigsberger, Leo. "The Investigations of Hermann von Helmholtz on
The Fundamental Principles of Mathematics and Mechanics." In Annual Report of the Smith
Inst to July 1986. Washington 1898. Pp.93-124. P.101.
66 See the dedication, afterwards withdrawn, on the manuscript of the Erhaltung:
Helmholtz, Hermann.Über die Erhaltung der Kraft . Christa Kirsten ed. Weinheim: Physik-
Verlag, 1983, P.14; see also Koenigsberger H v H Pp.35-7.

experiments in the most disparate branches of physiology and physics 67. This
claim is difficult to accept: actually the Erhaltung, as I am about to show, does
not have a specific experimental character, nor does it offer new experimental
results. The only possible interpretation of Koenigsberger's claim is to relate all
of Helmholtz's experimental activity of the period to the parallel researches on
heat and muscular action, that is to the preparation of his sixth paper.
The 23rd of July 1847 the Erhaltung was presented, with great success,
at the Physikalische Gesellschaft. However, Magnus' and Poggendorff's
judgements were not so warm and publication in the Annalen was denied. The
essay was finally published at Reimer.
Reimer's edition 68 of the Erhaltung consists of 1) an Introduction, of
methodological and philosophical character, and six chapters. The first two
67 Koenigsberger: ibid.P.37
68 Different editions and translations of the Erhaltung have been published: A) the
transcription of the penultimate manuscript: Helmholtz, Hermann.Über die Erhaltung der
Kraft . Christa Kirsten ed. Weinheim: Physik-Verlag, 1983; B) Reimer's edition: Helmholtz,
Hermann. Über die Erhaltung der Kraft, eine physikalische Abhandlung . Berlin: G.Reimer,
1847; C) the "historical" translation of Tyndall: Helmholtz, Hermann. "On the Conservation of
Force; a Physical Memoir." Trans.John Tyndall. In Scientific Memoirs-Natural Philosophy,
Tyndall and Francis (eds.), vol. I, p.II, London,1853:114-162; D) a French translation,
probably prompted by Clausius (see Wolff's "Clausius"): Helmholtz, Hermann. Mémoire sur la
conservation de la force précédé d'un exposé élémentaire de la transformation des forces
naturelles, trans L.Pérard. Paris: V.Masson et Fils, 1869; E) Helmholtz's own reprint in the W
A 1 of 1882. This edition includes some footnotes and six appendix; moreover the only final
note of Reimer's edition is included in the text: Helmholtz, Hermann. "Über die Erhaltung der
Kraft" in WA1 , 1882, pp.12-75; F) the reprint in Ostwald's series. Here the 1882 footnotes do
not appear: Helmholtz, Hermann. Über die Erhaltung der Kraft, eine physikalische
Abhandlung . In Ostwalds Klassiker der Exacten Wissenschaften . Nr.1. Leipzig:
W.Engelmann, 1889; G) an Italian translation, accurate but without the 1882 footnotes:
Helmholtz, Hermann. "Sulla Conservazione della Forza." In Opere . V.Cappelletti ed. Torino:
UTET, 1967 : 49-116; H) a new English translation, in my view controversial: Helmholtz,
Hermann. "The Conservation of Force: a Physical Memoir." In Selected Writings of Hermann
von Helmholtz . R.Kahl ed. Middletown: Wesleyan U.P., 1971: 3-55; I) a third English
translation, whiggish (Conservation of Energy ) but with comments: Helmholtz, Hermann.
"On the Conservation of Energy." In Applications of Energy: Nineteenth Century . R.Bruce
Lindsay ed. Stroudsburg: Dowden Hutchinson & Ross, 1976, Pp. 7-31 and 226-243.

chapters, 2)"The Principle of the Conservation of Living Force " and 3)"The
Principle of Conservation of Force", are dedicated to formulating the principle;
the following four chapters are dedicated to the applications of the principle in
the fields of 4) Mechanics, 5) Thermology, 6) Electrostatics and Galvanism, 7)
Magnetism and Electromagnetism, respectively.
1 The "Einleitung": intelligibility of nature and conceptual explanation
In February 1847 Helmholtz sent a sketch of the Erhaltung 's Introduction
to DuBois: it was immediately praised as "an historical document of great
scientific import for all time" 69. This apparently over-enthusiastic judgement
proved to be correct: almost fifty years later Helmholtz was to apply the same
concepts in the Introduction to his series of Lectures in Theoretical Physics ,
ninety years later Einstein and Infeld quoted it 70 as the paradigm of the
mechanical conception of nature, and today it still is the object of methodological
debates 71.
The history of the Introduction is complicated: when the Erhaltung was
presented the 23rd of July at the Physikalische Gesellschaft and when sent to
Magnus in the hope of publication in the Poggendorff's Annalen, the Introduction
had been dropped. After Poggendorff's refusal, at DuBois' request, it was restored
altered in "certain parts" 72 when the essay was sent to Reimer to be published.
In my view these modifications can be identified with the addition of what
is now the first paragraph of the Introduction: they represent an opening
paragraph in which the plan of the Erhaltung is sharply summarised, and are of
extraordinary relevance for an understanding of Helmholtz's approach.
69 Koenigsberger : H v H P.37.
70 Einstein, Albert and Infeld, Leopold. The Evolution of Physics. New York: Simon
& Schuster, 1938; chapt.1, sect 9: The philosophical background.
71 Turner, Steven. "Hermann von Helmholtz." In DSB 6, 1973. Pp.241-53; Heimann,
Peter. "Helmholtz and Kant: The Metaphysical Foundations of 'Über die Erhaltung der Kraft'
." In SHPS 5 (1974) : 205-38; Galaty, David. "The Philosophical Basis of Mid-19th Century
German Reductionism". In Journal for the History of Medicine and Allied Sciences 29
(1974): 295-316; Cohen,Robert and Elkana, Yehuda. "Introduction". In H.v.Helmholtz .
Epistemological Writings . Boston: Reidel, 1977. Pp.IX-XXVIII. .
72 Koenigsberger : H v H. P.38.

The premise reveals that the structure of the Erhaltung is based on four
relevant methodological layers 73: a) to establish two physical assumptions
("physikalischen Voraussetzung": central Newtonian forces and impossibility of
perpetual motion) and their equivalence 74; b) to derive from them as a
consequence ("Folgerungen") a theoretical law ("die Herleitung der aufgestellten
Sätze": the principle of conservation of energy) 75; c) to compare this general
principle with the empirical laws ("erfährungsmässigen Gesetzen") which connect
the d) natural phenomena ("Naturerscheinungen") in various fields of physics 76.
Helmholtz thus not only plans to offer, at variance with most of the other
researchers involved with conservation problems, a specific functional
formulation 77 of the quantities conserved and of their interrelations, but also a
derivation of this "principle" from more general physical assumptions. This is an
implicit assertion of the possibility of alternative versions of the principle.
But the great theoretical innovation is that empirical laws are supposed to
be compared no longer only with natural phenomena, but also with a general
principle. It is not difficult to understand Magnus' and Poggendorff's perplexities
in the evaluation of the essay: the young physiologist without presenting new
experimental results adds two levels (a, b) to the standard practice of
(experimental) physicists - that of formulating empirical laws (c) which would fit
natural phenomena (d).
One of the first conscious criteria of demarcation between theoretical and
empirical science can now be drawn 78: while the experimental scientist is looking
for empirical generalisations that fit experimental data (e.g.: the refraction and
reflection laws), the theoretical scientist looks for the agreement of the principle
of conservation with existing empirical laws (justificatory role of the principle)
and for the theoretical discovery of new ones (heuristic role). Helmholtz here is
explicitly setting out the task of theoretical research for the following decades:
agreement with principles will become a condition which empirical laws have to
satisfy, as important as the agreement with experimental data.
73 Helmholtz Erhaltung P.1
74 To be done in section 1
75 In section 2
76 In sections 3, 4, 5, 6.
77 A general framework that allowed energy terms, while retaining some constant
features, to acquire diferent expressions in different applications.
78 Helmholtz Erhaltung P.2

The methodological skills of Helmholtz span on an even broader horizon:
he believes that this already sophisticated approach has to be justified on
philosophical grounds. Helmholtz was clever enough to realise that this would
have been too much for contemporary physicists, and, as already recounted,
initially dropped the Introduction, unfortunately without achieving the goal of
publication. The actual scope of the Einleitung, later restored with the addition of
the premise, is to add an extra level to this already sophisticated pyramid of
knowledge. In fact it was meant to show the meaning of the two initial
assumptions for the final ("letzten") and true ("eigentlichen") goal ("Zweck") of
physical sciences 79.
Helmholtz believes that this means: a) to find the unknown causes of the
phenomena, and b) to understand them through the law of causality 80. While the
unknown causes ("unbekannten Ursachen") are going to be explicitly identified
with constant Newtonian forces 81, what is meant here by law of causality
("Gesetze der Causalität") is more difficult to explain. My interpretation is that
here Helmholtz refers to the theoretical search of the "empirical" link between
natural phenomena and particularly to the causal link he is about to establish in
chapter 2 between living and tension forces. Thus in my view the law of causality
is taken here in its "regulative-empirical" sense 82, that is as a theoretical relation
between "empirical" terms.
But Helmholtz rapidly shifts to a different meaning of causality, a
"transcendental" one, that is, causality seen as a precondition for the possibility
of scientific knowledge: the scientist must assume that nature be intelligible, that
"every transformation in nature must have a sufficient cause ( jede Veränderung
in der Natur eine zureichende Ursache haben müsse)" 83. A natural process is
79 Helmholtz Erhaltung Pp.1-2.
80 Helmholtz in 1882 added here a note on causality and Kantianism; see below.
81 Interesting remarks on Helmholtz's relationship between law and force as an
antifunctionalist tool are in Lenoir:Strategy of Life at p.232; but Helmholtz did not assert "the
conservation and interconvertibility of all central forces" P.231.
82 For the distinction between "regulative-empirical" and "transcendental" causality in
Kant see: Buchdahl, Gerd. "Reduction-Realization: A Key to the Structure of Kant's
Thought". In Philosophical Topics XII Vol.2 1981-2: 39-98; specially Pp.83-4; Peter
Heimann: "Helmholtz and Kant".Pp.221-3; in n.54 Heimann refers to (older works of)
G.Buchdahl.
83 Helmholtz Erhaltung P.2

intelligible if it can be referred to final causes. In fact final causes act following a
constant law and thus, if the external conditions are the same (ceteris paribus),
they produce the same effect.
Helmholtz, probably in relation to the vital force debate, asserts that it
might be possible that not all natural processes are actually intelligible 84. Some
phenomena might belong to a realm of spontaneity and freedom, this cannot be
decided conclusively, but the scientist must, all the same, assume the
intelligibility of nature as the departure point for his investigations. Here, in my
view, comes into play the second basic physical assumption to be justified, the
impossibility of perpetual motion. The impossibility of perpetually providing
work without a corresponding compensation is in fact a limit to the spontaneity
and freedom of nature and a physical version of the principle of sufficient cause.
Often the Kantian character of the Einleitung has been stressed 85.
However it has to be remarked that different parts of Kant's work play different
roles here. Up to now Helmholtz has dealt with i) the regulative principle of
empirical causality, ii) the transcendental principle of causality as the one
granting the possibility of scientific knowledge and the lawlikeness of nature.
Instead, Helmholtz's preoccupation in the following pages is iii) the conceptual
explanation of a specific physical model, tending to show the possibility of
Newtonian forces and not, at this stage, their inductive validity 86, The specific
model is the Newtonian one, based on central forces depending only on distance.
Through a detailed conceptual explanation based on the mechanistic categories of
matter and force, Helmholtz tries to show that Newtonian forces can be
considered the ultimate causes of natural phenomena and follows the method of
Kant's Metaphysical Foundations of Natural Science . Helmholtz's famous
84 Helmholtz Erhaltung P.2
85 Helmholtz himself in 1882: W.A. 1 "Erhaltung" P.68; Ellington, J.W. "Kant". In
C.C.Gillespie (ed.) DSB VII New York: Scribner, 1973. Pp.224-35; P.234; Elkana, Yehuda.
"Helmholtz's 'Kraft': An Illustration of Concepts in Flux". In HSPS 2 (1970): 263-98;
P.Heimann: "Helmh and Kant"; Wise in: Wise, Norton. "German Concepts of Force, Energy
and the Electromagnetic Ether: 1845-1880" In Conceptions of Ether. G.N. Cantor and
M.J.S.Hodge eds. Cambridge : Cambridge U.P., 1981. Pp.269-307, agrees with the influence
of Kant's Metaphysical Foundations on Helmholtz, stressing the role of intensity and capacity
factors; Fullinwider, S.P. "Hermann Von Helmholtz: The Problem of Kantian Influence". In
Stud. Hist. Phil. Sci. 21 (1990): 41-55.
86 see Heimann: "Helmholtz and Kant" P.229.

enunciation of the mechanical world 87 view is based on the assumption that both
matter and force are abstractions and that the first cannot be considered more
"real" than the second. Here he makes explicit that the problem of finding
unchanging basic causes can be interpreted as the problem of finding constant
forces. Causes and forces can be identified. A characteristic of the definition of
force is that it is constant in time; bodies with constant forces only allow spatial
movements and if the forces of extended bodies are decomposed into forces
acting between material points, the intensity of forces depends only on the
distances. This in Helmholtz's view is a direct consequence of the principle of
sufficient reason ("Satz von zureichenden Grunde") 88. Thus if all natural
phenomena could be reduced, through a general application of the principle of
conservation, to the effects of attractive or repulsive forces whose intensities
depend only on distance, then "empirical" causality would match "transcendental"
causality and the task of physical science would be satisfied: an "intelligible"
nature would be "understood".
Helmholtz's conceptual model of forces was to be of special relevance in
the whole energy debate. It was by no means universally accepted. On the
contrary, Weber's electrodynamic law of 1846, based on the alternative
assumption of forces depending on distances, velocities and accelerations, was
gaining universal recognition. This might explain the particular care given in the
Introduction to the conceptual explanation of the model. No confusion is made
here between Kraft as energy and Kraft as Newtonian force: only the second
meaning is applied 89.
Helmholtz admitted that in the development of Mechanics the model of
the forces had not yet been limited to the Newtonian one, but claimed that this
was due to a lack of understanding of the foundations themselves of Mechanics.
Helmholtz claimed moreover that some of the most important mechanical
principles are valid only on the assumption of central forces depending on a
distance. Among these the principle of living forces plays a special role
(Helmholtz will "show" that the principle of virtual velocities can be deduced
from the one of vis viva 90) and thus it can be considered the most general and
87 See Einstein and Infeld quoted n.164.
88 Helmholtz Erhaltung P.5.
89 Heimann, contra Elkana, shows that the meaning of "Kraft" is always unequivocal
in the context: "Helmholtz and Kant" P. 207 n.10 and P.209 n.14.
90 Helmholtz Erhaltung Pp.6 and 17-8.

elevant consequence of the deductions made (the great role that Helmholtz gives
to the principle of vis viva will appear evident in the first chapter of the
Erhaltung ).
Helmholtz in 1882 added some appendices to the reprint of the
Erhaltung; two of which are relevant for the present discussion of the
Introduction. In the first 91 he asserted he is now (1881-2) less Kantian than in
1847. But he also reasserted that the causality principle is nothing else than the
presupposition of the lawlikeness of natural phenomena and reasserted the
identification of cause, force and law. In my view, after the controversies with
the metaphysicians and his stress on the empirical aspects of geometry,
Helmholtz wanted to detach himself from a rigid interpretation of the validity
and reality of Kant's categories, while reasserting their use, in particular causality,
as presuppositions. That is, with the terminology adopted here: detach himself
from the interpretation of Newtonian forces as final causes but still adhere to the
regulative and transcendental use of causality 92. Also the other Appendix show
that, after the electrodynamic debate of the seventies, the Newtonian force model
was shaken. I rather believe that Helmholtz wanted to detach himself from the
Kantian model of Newtonian forces and instead stress the Kantian belief in the
transcendental causality as condition for the possibility of experience, a lasting
methodological tool for Helmholtz till the "Introduction to Theoretical lectures"
of 1894.
In the second Appendix 93 Helmholtz again tackled the problem raised by
his model of central forces and reasserted, against criticisms, some aspects of his
conceptual explanation: the discussions of chapter 1 and 2 of the Erhaltung can
still be considered "partially" valid only if it is accepted that forces can be
decomposed into point forces, and that the principle of superposition holds. But
this last assumption has to be explicitly admitted and cannot be considered any
longer a necessary consequence of the intelligibility of nature. Finally he
91 Helmholtz WA1 P.68.
92 Kahl Selected P. 49, Lindsay Applications P.27, Galaty "German Reductionism",
and Fullinwider "Influence" P.53 wrongly interpret "stärker..., als ich jetzt" as "strongly, as I
still" instead of "more strongly, than I now" as more correctly Heimann does: "Helmholtz and
Kant" P.219. But, while agreeing with Heimann that despite qualifications the appendix still
reveals a Kantian framing, I do not believe that it was meant to "provide a stronger
justificational foundation for the central forces principles". Ibid p.220.
93 Helmholtz WA1 Pp.68-70.

mentioned the electrodynamic theories of Weber and Clausius that accepted
forces depending not only on distances but also on velocities and accelerations.
These theories are criticised: they contradict the principle of action and reaction
and Helmholtz's formulation of energy conservation. Moreover, in the case of
Clausius' law, the role played by empty space, if accepted, would imply the
abandonment of the goal of intelligibility: accepting something that acts but on
which we cannot act would mean "a complete renunciation of the hope of
solving completely the tasks of natural sciences" 94. Obviously a theoretical
criticism.
Thus, in 1882, Helmholtz showed he still adhered to the regulative and
transcendental use of causality but had problems with the conceptual model so
carefully outlined in 1847. Indeed the model, while being essential for his specific
formulation of energy conservation had been and still was the main target of
criticisms.
2 The two roots of the vis viva principle and their supposed equivalence
In the first chapter Helmholtz tried to demonstrate the equivalence of his
two basic assumptions (central forces and impossibility of perpetual motion)
through an analysis of the vis viva principle. His argument starts with the
enunciation of the principle of impossibility of perpetual motion:
"We will start moving from the assumption that it is impossible, through
any combination of natural bodies, to produce continually motive force from
nothing" 95.
Helmholtz asserted that Carnot and Clapeyron 96 had theoretically
deduced a number of laws from this principle and that his own aim is to introduce
the principle into every branch of physics "in the same way", to show both its
justificatory (that is, its applicability to all the cases where the laws of the
phenomena have been already determined) and its heuristic role (in offering a
guiding thread to the experiments).
94 Helmholtz WA 1 P.70.
95 Helmholtz Erhaltung P.7.
96 Already quoted in the "Bericht".

Needless to say the plan was to be carefully carried out, but not without
the introduction of subtle innovations of extraordinary relevance, which are not
limited to overcoming the caloric model for heat still adopted by Carnot and
Clapeyron. The assertion "in the same way" refers to the methodology, but by
no means indicates that the same expression of the principle was to be applied.
Helmholtz in fact reformulates the principle of impossibility of perpetual motion,
utilising the term "work" ("Arbeit"), and mechanical terms as "force" and
"velocity":
"...the quantity of work obtained when a system of bodies moves from
one position to another under the action of specific forces must be the same as
that needed to carry the system back to the original position, independent of the
way, the trajectory or the velocity of the change" 97.
Thus, the term "Arbeit" is now a function of the state (position) of the
system, it is a total differential : in a closed path work cannot be created, but
cannot be destroyed. This is a first great innovation. The new concept of work,
being now a function of the positions, can be equated to another function of the
position: the vis viva.
In fact from Galileo's relation
v=√2gh,
where v is the final velocity acquired by a body of mass m in a fall from
height h under the acceleration g, we derive that
the work mgh equates the expression
,
which is a function of the position too. Here Helmholtz uses again the
word "Arbeit" for work, and following explicitly the French engineers' definition
of"travail", in equating work and vis viva gives priority to the concept of work. In
fact he defines
and not as the measure of vis viva; in this way it
"becomes identical with the quantity of work" 98.
97 Helmholtz: Erh .P.8
98 Compare this awareness of the role of the work concept with the difficulties in
chapter 5 to adopt a definition of "potential in itself" equivalent to work. An indication of
Helmholtz's theoretical rather than mathematical approach. See below.

This is done in order
"to establish a better agreement with the customary way today of
measuring the intensity of forces". 99
Given the equivalence of work and vis viva the "mathematical expression"
of the principle of impossibility of perpetual motion is obtained, that is, the law
of the conservation of vis viva:
"When any number whatsoever of mobile point masses moves solely
under the influence of forces, which they exercise on each other, or that are
produced by fixed centres, then the sum of the living forces of all the point
masses together is the same at every instant of time at which all the points are in
the same relative positions respect to each other and towards possible fixed
centres, whichever their trajectories and their velocities during the time
interval" 100.
It is relevant to stress the specific meaning of "conservation" utilised: here
the quantity conserved (vis viva) is conserved at specific positions and not during
the process, a definition that echoes Huygens' results for the compound pendulum
and Lagrange's definition of vis viva conservation.
Helmholtz moreover wanted to show that the principle holds only if the
forces can be decomposed into forces of mass points that are central. From 101:
,
where q is the velocity of a mass point m moving under the forces
exerted by a fixed system A and x,y,z are the Cartesian coordinates, and from
where X,Y,Z are the component of the acting forces and dq=Xdt/m,
Helmholtz derived incorrectly ( by equating the corresponding components of the
second members):
99 Helmholtz : Erh. P.9. Helmholtz did not rederive the definition by himself as
instead suggested by Kuhn: Sim Disc p.88.
100 Helmholtz : Erh. P.9; the formulation was to be criticised in 1854 by Clausius for
whom the positions cannot be considered as "relative" but are actually referred to fixed
centres.
101 Helmholtz: ibid.P.11.

and from here that the direction and magnitude of the force must be
function of the position of m and thus of its distance from the attracting point a.
Helmholtz synthesized in this chapter many elements deriving from
different traditions and introduced many novelties sometimes implicitly and not
always with success.
The first remark to be made concerns the formulation of the impossibility
of perpetual motion: Helmholtz introduced specific mechanical concepts (work,
velocity, force) that did not belong to the Carnot-Clapeyron expression. There is
thus an attempt at framing the impossibility of perpetual motion in a mechanical
world view 102: an (implicit) step in the methodological strategy tending to show
that the two initial assumptions belong to the same conceptual scheme.
My second remark deals with the relevant innovation introduced by
Helmholtz, that is the interpretation of the term work ("Arbeit") as a total
differential in the (new) expression of the impossibility of perpetual motion. Here
Helmholtz unified two different traditions in mechanics, analytical mechanics
and mechanical engineering 103, and an old philosophical principle, partially
already recalled in the physiological discussions of the "Bericht": "nothing comes
out of nothing and nothing is destroyed" 104.
In the French tradition of mechanical engineering, which, as seen, was
well known to Helmholtz, the term "travail" had received full importance: the
principle of conservation of vis viva became the principle of transmission of
work 105; but while accepting the impossibility of creating work the French
engineers did not exclude that work could be lost 106. Being mostly concerned
102 Helm's criticism on this point is very sharp: he summarised the mechanical
hypothesis with the statement that "all the events can be explained in terms of forces that
originate accelerations". He warned that with Helmholtz's reformulation of the principle of
impossibility of perpetual motion and of the subsequent expression of the vis viva principle,
whenever the last one is applied we will be "obliged to imagine forces and velocities involved
in every experimental application, for instance in electrical or thermal phenomena". For Helm
the two roots of Helmholtz's Erhaltung are basically different and the one based on
Newtonian forces has to be rejected, together with the mechanical formulation of energy.
Helm Energetik P.41.
103 for a classification see: Grattan-Guinness "The varieties of mechanics"
104 WA 1 P.6
105 Rühlmann Maschinenlehre ; Haas Entwickl. Pp.73-83.
106 Haas: ibidem P.81.

with impacts, for them work was not a total differential and the concept of
potential, which was being developed, not only in France 107, in the tradition of
analytical mechanics, was not generally admitted 108. On the other side in the
analytical tradition, up to a certain date, the quantity that is now called potential
was not meant to be work stored in the system at a certain position, but, despite
formal equivalence, was understood only as a mathematical function of the
positions from which the forces could be derived. Force by displacement in the
direction of force was in this tradition a total differential but did not receive a
physical interpretation. Helmholtz very subtly and skilfully here unified the two
approaches, that is, the concept of work with the function of positions (but not
without problems in defining potential 109). Now work cannot be created and
cannot be destroyed, it is a state function (of the positions). Discussions in the
1880's outlined clearly that with this new "representation" of the principle of
impossibility of perpetual motion a relevant modification had been achieved 110.
The relevance of the innovation just introduced was made explicit by
Helmholtz himself in 1884 111; he asserted that there are two philosophical roots
of the principle of conservation: the "ex nihilo nil fieri" and the "nil fieri ad
nihilum". The first is connected with the impossibility of creating work ( and thus
with the impossibility of perpetual motion) and the second with the impossibility
of destroying it. Helmholtz in 1884, looking back, asserted that a big difficulty
he had to overcome in the formulation of energy conservation was the acceptance
of the "ad nihilum", while the first root was part of shared knowledge 112. The "ex
nihilo" is based on an inductive assumption, already made by a qualified minority
nn.71-4.
107 see for instance the contributions of Green, Hamilton, Gauss, F.Neumann: above
108 Grattan-Guinness, Ivor. "Work for the Workers: Advances in Engineering
Mechanics and Instruction in France, 1800-1830". In Annals of Science 41 (1984):1-33. P.
32.
109 See chapter 5 of the Erhaltung
110 Planck Princip P. 37.
111 In an appendix to the reprint of the famous talk on the Interaction of Natural
Forces: Helmholtz, Hermann. "Robert Mayer's Priorität". In Vorträge und Reden. 2 vols.
Braunschweig: Vieweg, 1884.
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
experiences aimed at the destruction rather than the production of work (and this
can explain the above mentioned approach of the French engineers) 114.
That Helmholtz apart from being aware of the French theoretical
engineering tradition was also fully aware of the tradition of analytical
mechanics can also be seen 115 in the similarities between his definition of the
principle of vis viva conservation and the one of Lagrange. Moreover the
"derivation" itself of this principle through Galileo's free fall law echoes the
historical account of the development of the principle of vis viva conservation
given in Lagrange's Mechanique Analitique
The latter's account 116 can be summarised in the following steps:
Descartes had defined work as force (weight) by displacement; from the
impossibility of perpetual motion Leibniz showed that vis viva ( ) is the "true"
measure of the capacity of doing work of a body in motion; Huygens through his
compound pendulum obtained the result that the sum of the vis viva of a system
of bodies is a function of the positions and is independent of the constraints of the
system and of the way in which the system of bodies reverts back to those
positions ( here of course the velocity v is the final velocity that a body in a given
position would acquire when "falling" from that position to a second position
assumed as reference). D.Bernoulli, utilising Galileo's free fall relation, stated
that the vis viva is equal to a product of force times a distance. D'Alembert and
Euler clarified the analytical relations between vis viva and work discarding
Leibniz's philosophical ideas on causality and utilising Newton's definition of
113 See the interesting pages of Leonardo, Cardano, Stevin and Galileo in: Lindsay,
Robert Bruce. Energy, Historical Development of the Concept. Stroudsburg: Dowden
Hutchinson and Ross, 1975.
114 Planck in 1887 gave great relevance to the two roots (chapt.1) and showed that
the principle of conservation of energy cannot be deduced from the impossibility of perpetual
motion only but that also the "ad nihilum" is needed (chapt.2). Planck Princip P.3, Pp.138-9.
Helm in 1898 still underlined the need for the two roots and quoted Planck's demonstration:
Energetik P.51.
115 apart from always dubious agiographical claims: Koenigsbeger: H v H P.30 (for
Helmholtz's knowledge of Jacobi's Funct.Ellipticarum ) and P.43 (knowledge of D.Bernoulli
and D'Alembert; also: Jacobi recognised links between the Erhaltung and the analytical
tradition).
116 Lagrange, J.L. Mechanique Analitique , Paris: Desaint, 1788.

force. Lagrange himself in the Mechanique Analitique from the principle of
virtual work deduced the vis viva principle outlining that the name conservation
of vis viva is given because the sum of the vis viva expressions of the bodies of
the system is the same when the bodies revert back to their original positions,
independently of the trajectories followed. Thus, for Lagrange, conservation of
vis viva is, after Huygens, conservation at specific positions 117 and not
conservation during a process. In Lagrange's formulation 118:
the stress is on the vis viva term: the factor 2 is with the second member;
the function of the positions (the potential term) and the constant F are not
given special relevance and do not acquire physical meaning, despite the fact
that, from a contemporary point of view, the two terms on the right side of the
equation are easily identified with total energy and potential energy of the
system 119.
My third remark outlines the criticisms to Helmholtz's "demonstration"of
the equivalence of the two initial assumptions. This "wrong" demonstration plays
a vital role in Helmholtz's research program: the generalisation of the principle of
conservation of vis viva into Helmholtz's principle of conservation of "force", that
is, into a principle where the kinetic and positional terms can be sharply split, is
only possible if the forces depend solely on distances. While this applies to
Newton's, Coulomb's and Ampere's forces, it does not apply to Weber's
electrodynamic ones, not a minor problem for Helmholtz. It being impossible at
the time to oppose Weber's law on empirical grounds 120, it was important for
Helmholtz to demonstrate that empirical laws which admitted forces different
from the central ones violated the conservation of vis viva and the impossibility
of perpetual motion.
117 This particular meaning of conservation surprises some commentators: Lindsay
(unhistorically) wonders why the name of conservation of vis viva is given if vis viva is
conserved only in the specific case of constant potential energy: Hist Devel Pp. 167-9.
118 Lagrange Mech An P.208.
119 The same can be said of other expressions of vis viva conservation typical of
analytical mechanics: P.S.Laplace, Pierre Simon.Traité de la Mécanique Celeste . 5 vols.
Paris, 1799-1825. Vol I, P. 52; Hamilton "On a General Method in Dynamics" Phil Tr Roy
Soc Pt II 1834: 247-57. P.250; in: Lindsay Hist P.264.
120 Helmholtz will try that in the seventies.

But unfortunately for the whole strategy this was an impossible task.
Helmholtz's demonstration was based on a wrong assumption: even if the
components of the vis viva depend on the positions only, the same does not
necessarily follow for the force components. In fact it was possible to show that
forces depending on velocities and accelerations, like Weber's, do not violate the
conservation of vis viva or the impossibility of perpetual motion: Weber in 1848
was to show that his own force admitted a potential, even if a kinetic one 121. But
all the same Helmholtz's approach had for twenty years an incredible success: in
the British literature the point that Weber's force law denied conservation of vis
viva and energy conservation was maintained by Maxwell 122 in 1865, Tait and
Thomson in 1867 123 and Tait in 1868 124, and finally refused by Maxwell in 1873
only after Helmholtz's own retreat in 1870. Helmholtz himself acknowledged in
1882 125 that a flaw in the 1847 demonstration had been shown by Lipschitz (he
still did not mention Clausius' criticisms 126). Nevertheless he had to agree that he
121 The first to explicitly criticise the limitations of the validity of the principle of vis
viva to central forces was Clausius in 1852 and 1854.
122 Maxwell, James Clerk. "A Dynamical Theory of the Electromagnetic Field". Repr.
in: Scientific papers. 2 Vols. Cambridge: Cambridge U.P., 1890. Vol.1 Pp.526-597; at
pp.526-7.
123 Thomson,W. and Tait, P.G. Treatise on Natural Philosophy. , Clarendon Press,
Oxford, 1867. Par. 385. P.311-2.
124 Tait,P.G. Sketch of Thermodynamics., Edmonston and Douglas, Edinburgh, 1868.
P.76. See also C.Neumann's defense of Weber : Neumann, Carl. Die Gesetze von Ampére und
Weber. Leipzig: Teubner, 1877. Pp.322-4; for a discussion see my Energy and Electr Pp.122-
3 and133-6.
125 Helmholtz WA 1 P.70.
126 Clausius, Rudolf. "Ueber das mechanische Aequivalent einer elektrischen
Entladung und die dabei stattfindende Erwärmung des Leitungsdrahtes" In Pogg
Ann 86 (1852): 337-375
Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber
die Erhaltung der Kraft." In Pogg Ann 89 (1853): 568-579.
Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber
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
privileged status 127.
Helmholtz's foundations of the Erhaltung are wide ranging but hide a not
insignificant weakness: the equivalence between the two conceptual roots
(impossibility of perpetual motion and the resulting conservation of vis viva on
one side, and the forces depending only on the distance on the other) had not
been and could not be demonstrated. This was to become explicit very soon.
Based thus on unsecure deductive grounds the validity of Helmholtz's approach
had to rely on the inductive side: on its "empirical" success; that is on its
capability of reassessing already existing knowledge (justificatory power) and on
the capability of disclosing the interpretation of new phenomena (heuristic
power).
3 The duck, the rabbit and the principle of energy conservation.
Helmholtz was now ready for a great generalisation: in the second chapter
the principle of conservation of vis viva will become the principle of conservation
of "force".
127 As already recalled Helmholtz still in 1882 tried to save his original approach with
the introduction of a new hypothesis: the forces that do not obey a central Newtonian force
law cannot fulfill the action and reaction principle. Helm in 1898 restated Clausius' criticisms
but from a different perspective: Clausius shared the mechanical point of view and attacked the
privilege given to the Newtonian forces, Helm instead criticised the mechanical world view
and the supposed equivalence of the two roots of Helmholtz: the flaw in the demonstration
shows, according to Helm, the independence of the principle of perpetual motion from the
model of Newtonian central forces. A relevant remark is shared by Planck and Helm:
Helmholtz, together with the impossibility of perpetual motion, made implicit use of another
principle to avoid the problems created for his own approach by the irreversibility of many
transformations in nature: the principle that a state, derived from another, can always be
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 with
velocity q, along the path r, under the action of a central force , can be written
as:
or if Q and q are the velocities at the distances R and r:
128
This is formally identical with the well known theorem of vis viva-work.
The first term is in fact the well known variation of the vis viva and the second
term has the dimension of work (force by elementary displacement in the
direction of the force integrated along a line). In the first chapter Helmholtz often
utilized the word "Arbeit" so we would expect it mentioned again here. Instead a
bold reinterpretation of the equation takes place, the second member of the
equation is in fact not defined as "Arbeit" but as:
"the sum of tension forces ( Spannkräfte) between the distances R and
r". 129
Helmholtz made an effort at clarifying the innovation: the tension force
was meant to be in explicit conceptual duality with the living force ("in contrast
to that which Mechanics calls living forces" 130), a "force" that tends to move the
point m, until movement actually takes place. A geometrical interpretation of the
concept is given, it represents: "the set of all the intensities of the force acting in
the distances between R and r". In fact if the intensities of correspond to
ordinates perpendicular to the line of abscissae connecting the point m and the
centre of force a, the integral represent an area given by the "sum of the infinite
abscissae (read: ordinates) lying on it".
The partly unsuccessful effort (the integral is not the sum of the abscissae
but of infinitesimal surfaces) to give a geometrical interpretation of the concept of
Spannkraft, stresses the fact that the tension forces dr are also very different
from the Newtonian forces : dimensionally they are represented by the product
of a force by a displacement; they exist when the material point is not in motion,
tend to put it in motion and are "consumed" by the acquired motion (compare
with the constant relation force-matter described as conceptual model for the
128 Helmholtz Erhaltung p.13
129 Helmholtz Erhaltung p.14
130 Ibid

forces in the Introduction); while they are function of a distance (two positions)
they acquire a proper meaning only when summed over a definite interval.
Helmholtz introduced relevant new theoretical concepts in an old
equation. Now both the first and second member have a physical theoretical
meaning, connected by an equality that does hold during a process: a variation of
one member equates the variation of the second member. The sum of the two
members is physically interpreted as a third theoretical concept, later to be called
energy:
" 'In all cases in which free material points move under the influence of
attractive and repulsive forces: forces, whose intensity depends only on the
distance, the loss in quantity of tension force is always equal to the gain in living
force, and the gain in the first is always equal to the loss of the second. Therefore
the sum of the existing tension and living forces is always constant.' In this quite
general form we can define our law as the principle of conservation of force." 131
In the case of Huygens' conservation of vis viva, the meaning of
conservation itself was different: conservation meant that the vis viva of a system
reacquires the same value when the system reacquires the same positions,
independently of the trajectories to revert back to those positions; of course
velocity, and thus vis viva, changes during motion.
Helmholtz's conservation instead introduced a new meaning of
conservation: Kraft (energy) is conserved during motion and a variation of vis
viva corresponds to an opposite variation of tension force.
Finally Helmholtz presented the deduction of the principle of virtual
velocities from the conservation of force 132 mentioned at the end of the
Introduction: an increase of vis viva can result only from the consumption of a
quantity of tension force. Thus a system at rest, if there is no consumption of
tension forces for every possible direction of motion in the first instant, stays at
rest.
Helmholtz summarized the results so far obtained: a) the principle of
conservation of force implies that the maximum quantity of work that can be
obtained from a system is a determined, finite quantity if the acting forces do not
depend on time and velocities ; b) if they do or if the forces act in directions
131 Helmholtz Erhaltung p.17
132 Clausius was to invert the priority and deduce the conservation of vis viva from
the virtual velocities principle, more in line with the analytical tradition, in his 1852 paper and
in the various editions of his Potential and potential function.

different from the one joining the active material points "force" could be gained
or lost ad infinitum; c) if forces were different from central, a system of bodies at
rest could be set in motion by the effect of its own internal forces 133. The
hypothesis of central forces depending only on distances is thus basic to
Helmholtz's view, but these three results are not without problems as seen in the
previous subsection.
In my view Helmholtz's summary of his second chapter does not do
justice to the results he had obtained. A real shift in meaning occurred in fact: the
well-known old equation written at the beginning of the chapter had acquired a
new interpretation; we already knew that the stress could be with the tradition of
analytical mechanics on the first term (the duck : conservation of vis viva), with
mechanical engineering on the second term (the rabbit: transmission of work);
now with Helmholtz the stress was on the equivalence between the two.
The introduction of the term Spannkraft reveals a real meaning shift: with
the tension force we are very far from the concept of work and very close to the
idea of potential energy. It is not in fact work done but work that can be done,
capacity to do work. Work acquires now the role of a unit of measurement for a
new theoretical concept. Planck outlined the great importance of the step
undertaken:
"However insignificant this interpretation might seem at first glance, the
perspective that it opens on all fields of physics is nevertheless extraordinarily
wide, because now the generalisation to every natural phenomenon is evident." 134
In 1887, with prophetic insight, Planck declared 135 also that with
Helmholtz's formulation the principle of conservation of "force" became similar
to the one of conservation of matter: "force" as matter cannot be increased or
diminished, but can appear in different forms. The two basic forms of "force", vis
viva and tension force, can appear in many ways: vis viva as motion, light, heat;
tension force as elevation of a weight, elastic or electric potential, chemical
difference 136, and so on.
"But the sum of all these reserves of force (so to say accumulated in
different stores) is invariably the same and all natural processes only consist in
133 Helmholtz Erhaltung Pp.19-20
134 Planck Princip P.37.
135 Planck Princip P.37
136Probably in the sense of different energy levels connected with chemical bonds.

the simple passage from one to the other" 137. This was "another step forward
towards simplifying the understanding of all natural phenomena" 138.
It is indeed surprising that such a great innovation is seen as a failure by modern
commentators 139 and discarded; I believe instead that an analysis of its origin
outlines some specific features of a theoretical and a mathematical approach to
physics. This can be shown relating Helmholtz's approach with Leibniz's one and
comparing it with Clausius's one.
In my opinion, to achieve his result, Helmholtz made use of a Leibnizian
inheritance 140. First the duality lebendig Kraft-Spannkraft strongly resembles the
older one between vis viva-vis mortua; the terms themselves are very similar:
while the vis viva has almost the same expression, the meaning of Spannkraft is
very close to the older Leibnizian counterpart:
"the ..... vis viva, produced by an infinite number of applications of vis
mortua " 141
But between Helmholtz's and Leibniz's ideas of positional "energy" there
are two main differences that explain the success of the more recent formulation:
a) Helmholtz provided a formal quantitative expression; b) the Newtonian forces
are part of it.
Helmholtz, making proper use of the Newtonian concept of force, and
accepting the full inheritance of Newtonian mechanics, provided a formal
expression for the second term of the duality that was missing in Leibniz.
But other Leibnizian elements can be outlined: the equality of the two
terms in the mathematical expression is no longer the indication of an analytical
identity. Being two independent physical concepts, now the equality has the
meaning of a causal relation: the variation of one member implies the variation of
the other. The equality holds at every instant during a process. This is a
Leibnizian concept of conservation:
137 Planck Princp P.37
138 See Planck Princip P.37
139 Kuhn thinks that Helmholtz "fails to recognize" the integral as "Arbeit", see above
n.19; Lindsay too is surprised for the lack of the term "Arbeit" and stresses the mistake of
considering an integral as the sum of lines. Lindasy Applic. of Energy P. 16.
140 A Leibnizian influence on Mayer has been often remarked, on Helmholtz it has
been only briefly pointed out by Planck Princip P.35, Koenigsberger H v H P.49; Elkana
Disc Chapt 1 n.31.
141 Leibniz : Specimen Dynamicum (1695) quoted in Lindsay Hist P.122.

"The equality that exists between effect and efficient cause confirms what
we have just said. In this equality consists the conservation of forces of bodies
that are in motion" 142.
"Empirical" causality then comes into play here: not causality as a
condition of the possibility of natural laws, but a principle which establishes a
specific link between different realms of phenomena. The "empirical" causality
indicates a quantitative equivalence between phenomena that are qualitatively
different (static and dynamic).
A static cause can generate, as an effect, the movement of a body. This
movement, in its turn the cause, has the power ("motive force") to produce the
effect of bringing the body back to its former position. The quantitative
equivalence of cause and effect is supposed to hold true at every instant of the
process: the interchangeability of the initial and final stage is only an
exemplification of this principle. What is conserved during the process in
Helmholtz as in Leibniz is this specific quantitative equivalence between
qualitatively different phenomena.
But how can we actually measure two qualitatively different phenomena
to establish a quantitative equivalence? A common unit of measurement is
needed. Again a problem already faced by Leibniz, who asserted the relevance of
(what was later defined as) work as a unit of measurement of all natural
phenomena 143. Leibniz had recognised the impossibility of continually creating
work without a corresponding compensation, and also of destroying work, i.e. of
both the "ex nihilo" and "ad nihilum", a necessary condition to guarantee the
invariability of the chosen unit.
142 Johann Bernoulli Discours sur les lois de la communication du mouvement, Paris
1724; chapt.V, par. 8; quoted in Lindsay Hist P.125.
143 Leibniz:"Brevis Demonstratio erroris memorabilis Cartesii et aliorum circa legem
naturalem, secundam quam volunt a Deo eandem semper quantitatem motus conservari, qua et
in re mechanica abutuntur" in Acta Eruditorum, Christofory Guntheri, Lipsiae 1686, Pp.161-3;
repr. in Leibniz Mathematische Schriften, 7 volumes, C.I.Gerhard ed, Berlin: Asher and Halle:
Schmidt, 1849-63; Vol 2, 1860, Pp.117-9; more generally see: Cassirer, Ernst. Leibniz' System
in seinen wissenschaftlichen Grundlagen, Marburg: Elwert, 1902; Substanzbegriff und
Funktionsbegriff, Berlin: B.Cassirer, 1923, Chapt 4, Sect 7, pp.226-48; Das
Erkenntnisproblem in der Philosophie und Wissenschaft der Neueren Zeit , 2 vols. 2nd
revised edition, B.Cassirer: Berlin, 1911; vol 2. P.165.

In Helmholtz's essay too, work is the common unit of quantitative
measurement for different phenomena, phenomena connected by a causal
principle, but this time mechanically interpreted under Newton's definition of
force and laws of motion. The "ex nihilo" and "ad nihilum" are now both present:
the quantitative aspects of one member of the equation have to be the same as of
the other member, no more nor less. Work cannot be created, but neither
destroyed and thus a great generality is acquired.
Helmholtz's program is thus that all natural phenomena be measured by a
common unit and be interpreted through only two forms of "energy": without
doubt a great theoretical unification resulting from the above mentioned meaning
shift.
A deeper understanding of Helmholtz's result can be reached, in my opinion,
through a comparison with a less philosophically but more mathematically
inclined approach: the one of Clausius. This will be done in the next chapter.
Another interesting comparison is the one between Helmholtz and Mayer: It is
surprising that Helmholtz, despite being acquainted with Liebig's papers was not
aware of Mayer's 1842 contribution to the Annalen der Chemie und Pharmacie .
Nowhere in the Erhaltung Helmholtz quoted Mayer and he later claimed not to
have known in 1847 of his paper. Still it is relevant to point out some similarities
and differences between the two approaches : the use of a Leibnizian causality
principle was in common, but the mechanical conception of nature , the
mechanical theory of heat, the central force hypothesis and the reduction of all
the qualitatively different forms of "force" to two basic ones were specific of
Helmholtz. Mayer disagreed with all these elements and his expression of
conservation of energy was closer to a principle of equivalence or in other words
to a correlation principle. On the other side Mayer had worked out a mechanical
equivalent of heat, even if not through original experiments but definitely through
original thinking, while, as I will show in subsection D, Helmholtz had not 144.
144 On Mayer see: the Tyndall-Tait debate of 62-65; above n.3; Helmholtz's remarks
in the 1882 appendix to the Erhaltung : WA 1 Pp.71-3, Helmholtz, Hermann. "Robert Mayer's
Priorität". In Vorträge und Reden. 2 vols. Braunschweig: Vieweg, 1884; Helmholtz's appendix
to " Das Denken in der Medecin". In Vorträge und Reden. 2 Vols. Braunschweig: Vieweg,
1884; see also Planck Princip Pp.21-8; Helm Energetik Pp.16-28; Haas Entwickl Pp.61-2;
more recently: Lindsay, Robert Bruce. Julius Robert Mayer, Prophet of Energy. Oxford and
New York: Pergamon Press,1973; Heimann, Peter. "Mayer's Concept of 'Force': the 'Axis' of a
New Science of Physics." In HSPS 7 (1976) : 227-96.

In virtue of its limitations to Newtonian forces Helmholtz's "Erhaltung der Kraft"
achieved in the second chapter, with the sharp distinction between kinetic and
positional terms, is the fulfilment of the far- reaching plan anticipated in the first
lines of the "Einleitung" : the conservation law just formulated is the
"consequence" (level B) of the two basic physical assumptions (level A:
impossibility of perpetual motion and central forces).
But so far we only have a theoretical framework that has to be applied, i.e. filled
with the specific expressions of the vis viva and tension forces resulting from the
interaction of the principle (level B) with the experimental laws (level C) in the
various realms of natural phenomena (level D). This interaction between levels B
and C will be carried forward in the next four sections.
Nevertheless, Helmholtz's achievements in the Einleitung and in the first two
chapters are impressive. In a limited number of pages a complete reappraisal of
previous traditions in physics has been made, with an original synthesis of
different and previously competing approaches (analytical mechanics, Newtonian
mechanics, Leibnizian philosophy, mechanical engineering ). But the programme
is only at its initial stage : the general framework is going to be applied in the
following four sections to an extremely wide variety of natural phenomena and
empirical laws. It is in this way that the "empty" framework will show its
justificatory and heuristic power, but not without problems. In my view it is
useful to analyse in the following chapters of the Erhaltung the difficulties
Helmholtz had to face in specifying the "energy" equations in the different
branches of natural sciences. This in order to assess the merits and demerits of
his mechanical approach based on the central force assumption, with the resulting
distinction between the two main forms of energy. An evaluation of Helmholtz's
results in the application of the theoretical framework to the empirical laws of
natural phenomena is a kind of analysis well known at the turn of the century,
but forgotten in more recent times.
4 An easy start: Mechanics
In the third chapter of the Erhaltung Helmholtz started to apply the principle to
mechanical theorems, mostly recalling applications already known of vis viva
conservation. That is: in this short and non- mathematical chapter Helmholtz did

not deal with specific applications of the concept of tension forces. He
considered briefly: the motions caused by gravitation, both of celestial and
terrestrial bodies; the transmission of movements through incompressible solids
and fluids, if vis viva is not lost through friction or inelastic collisions; the
motions of perfectly elastic 145 solid and fluid bodies without internal friction 146. It
is to be noted here that, dealing with wave motions, Helmholtz asserted that
"likely" light and radiant heat have to be considered, together with liquids and
sounds, thus showing an early acceptance of Melloni's theory. 147.
The use made by Fresnel of vis viva conservation in deriving the laws of
reflection, refraction and polarisation of light is explicitly mentioned, together
with the application of the principle to interference, thus displaying a broad and
deep knowledge of physical problems and physical literature. 148
The application of the principle of "Kraft" suggests that if there is a loss of vis
viva due to absorption of elastic, acoustic or heat waves, a different kind of
quantitatively equivalent "force" must appear 149. Heat must be produced by the
absorption of caloric rays, but Helmholtz asserted that it has not yet been proved
experimentally that the same quantity of heat which disappears in the radiating
body reappears in the irradiated one (if losses are excluded) 150. This is a first
instance of Helmholtz's approach in the last four chapters of the Erhaltung : to
suggest and outline applications of the principle, independently of experimental
corroborations.
Energetik P.59.
145 Comments on Helmholtz's definition of elasticity and on interference in: Helm
146 Helmholtz Erhaltung Pp.20-1.
147 Planck Princip P.39 noted Helmholtz's early acceptance of this theory. Brush
showed that between 1830 and 1850 a number of physicists believed in a wave theory of heat,
based on the analogies discovered by Melloni between radiant heat and light. The influences
on the "energy" debates are discussed, including Helmholtz's explicit references in the
Erhaltung . Brush, Stephen. "The Wave Theory of Heat: A Forgotten Stage in the Transition
from the Caloric Theory to Thermodynamics". In BJHS 5 (1970-71): 145-67. Reprinted in:
Brush, Stephen. The Kind of Motion We Call Heat. 2vols. Amsterdam: North Holland, 1976.
Vol.2 pp. 303-25. .
148 Helmholtz Erhaltung P.23.
149 Helmholtz Erhaltung P.23.
150 Helmholtz Erhaltung P.24

Helmholtz, while asserting that light absorption can cause heat, light
(phosphorescence) and chemical effects, identified light with radiations
producing thermal and chemical effects 151.
The chapter ends with some remarks on the (small) effects of light and chemical
rays on the eye, an indication perhaps of a small value for their heat equivalent 152.
The quantitative relations of the chemical effects produced by light were not well
known, and Helmholtz believed that relevant magnitudes were only involved in
the case of light absorbed by the green parts of plants 153.
Even in this short chapter, which mostly recalls already known results,
Helmholtz's method started to reveal its fruitfulness in organising a very large
amount of physical knowledge. But some limits also became evident: the
difficulty of identifying the "tension forces" reduces the conservation of "force" to
a correlation principle here. Moreover, it is evident that the principle can only be
heuristically fruitful if its predictions are supported by already existing empirical
laws in the various fields referred to. Even in this case, for the corroboration of
the principle, the knowledge of specific coefficients of equivalence is necessary,
but Helmholtz lacked this knowledge. His efforts had thus to be confined to
broad theoretical applications based on his surprisingly deep knowledge of the
physical literature.
5 Force equivalent of heat: a theoretical approach
The fourth chapter of Helmholtz's Erhaltung deals with the force equivalent of
heat. It would be easy to believe this to be the central part of the essay, but in fact
it is not. It is, rather, the chapter in which most notably the differences with the
papers of Mayer (unknown to Helmholtz) and Joule (known) become evident.
Surprisingly, Helmholtz, at variance with the other two, did not establish with
accuracy the mechanical equivalent, and did not seem concerned by the problem.
He was more interested in a theoretical interpretation of the thermal phenomena
151 Helmholtz Erhaltung P.24. He quoted Melloni and Brucke, always with
references in Poggendorff's Annalen .
152 Helmholtz Erhaltung P.24.
153 Helmholtz Erhaltung P.25.

through his own framework than in a specific determination of the equivalent. In
this long chapter the use of the formalism is limited to a discussion of Clapeyron's
and Holtzmann's laws and the whole approach is rather qualitative. Strangely
enough, the results here will not be as far-reaching as in the next two chapters.
The lack of a determination of the mechanical equivalent in this fourth chapter
probably justifies Helmholtz's later claim 154 that the Erhaltung was meant more
as a review and a synthesis of contemporary physical knowledge than to produce
original experimental results.
Helmholtz started applying his method of looking for actual compensations
(equivalents) to an apparent loss of "force" (energy). Through his own principle
of conservation of force he identified the compensation for the loss of living
force in inelastic collision and in friction with a supposed increase of tension
forces, namely internal elastic forces, due to the variation of "the molecular
constitution of the bodies" 155 and with acoustical, thermal and electrical effects.
First comes the case of collision of non-elastic bodies: the vis viva lost must
reappear as an increase of elastic forces (tension forces), as heat and sound. In
the case of friction instead we have increase of elastic forces, heat and electricity.
If we do not take into account molecular effects and electricity, we can pose two
great problems: a) does a loss of vis viva correspond to an equivalent amount of
heat?; and then: b) how can we frame heat in a mechanical interpretation? 156 The
first question is connected to a "correlation" approach, the second one is specific
to Helmholtz's program.
Surprisingly the first question is disposed of very rapidly, in a few lines, and
without definite results.
Helmholtz, who did not know of Mayer's and Colding's research, asserted that to
this problem "perhaps" too few efforts have been dedicated. He actually only
cited a paper of Joule (1845) 157 and recalled his attempts at establishing a
mechanical equivalent through the heat produced by the friction of water in
narrow tubes and in vessels (the famous paddle wheel experiment). Helmholtz
asserted that Joule's result in the first case is that the heat needed to raise by 1°C
154 Helmholtz WA 1 P.74.
155 Helmholtz Erhaltung P.26.
156 Helmholtz Erhaltung P.27
157 Joule, James P. "On the Existence of an Equivalent Relation between Heat and the
Ordinary Forms of Mechanical Power." In Phil. Mag ser3, XXVII, p.205, reprinted in
Scientific Papers Pp.202-5 , signed August 6,1845.

the temperature of 1 Kg of water can raise 452 Kg to 1 meter, and 521 Kg in the
second case 158. Helmholtz's judgement on Joule's work, the only original
experimental determination of the mechanical equivalent of heat cited in the
Erhaltung, is very severe. For Helmholtz, Joule's measurements are not adequate
to the "difficulties of the research" and thus it is not possible to accept the results
as correct: "probably the figures are too high". Helmholtz's criticism of the only
empirical evidence corroborating his own theoretical approach is surprising.
Koenigsberger later asserted 159 that he had read Joule's papers just before
publishing his own essay, but nevertheless, being a good experimentalist and
being involved in experimental physiological research in a closely related subject,
the judgement of inaccuracy referred to experiments that were indeed very
accurate requires an explanation.
Tyndall, trying to provide such an explanation in 1853, in a note 160 to the English
translation of the Erhaltung , warned the reader that Helmholtz was only
acquainted with Joule's early works; Tyndall again in 1863 161, during the famous
controversy with Tait and Thomson on the Mayer or Joule priority, quoted
Helmholtz's remarks and to justify them asserted that Joule's measurements of the
equivalent in 1843 162 varied between 1040 and 587 foot pounds. But actually
Helmholtz, in the passage of the Erhaltung we are dealing with, did not refer to
Joule's 1843 but to his 1845 paper. In this paper Joule gave the results of the
paddle wheel experiments (890 foot pounds) and recalled his previous results 163:
823 fp. in 1843 from magnetoelectrical experiments, 795 fp. in 1845 from the
158 For a discussion of these figures see below.
159 Helmholtz "On the Interaction of Natural Forces and recent Physical Discoveries
bearing on the same" Phil Mag XI, 1856:489-518, on p.499; see also Helmholtz
Autobiographical Sketch P.12; Koenigsberger HvH P.44.
160 Helmholtz, Hermann. "On the Conservation of Force; a Physical Memoir."
Trans.John Tyndall. In Scientific Memoirs-Natural Philosophy , Tyndall and Francis (eds.),
vol. I, p.II, London,1853 :114-162; P.131.
161 Tyndall, John. Philosophical Magazine 1863 Pp. 375-6
162 Joule, James. "On the Caloric Effects... " 1843. Repr in The Scientific papers
Vol.1. Pp.123-159. Pp.151 and 153.
163 Joule "Equiv Relat" quoted n.244, P.204 repr.

arefaction of the air 164, 774 fp. from unpublished experiments on the friction of
water moving in narrow tubes 165. Joule averaged the two experiments resulting
from the friction of water (890 and 774) to 832 and again averaged the results of
the three distinct classes of experiments mentioned (823,795 and 832) to 817.
The results are then far more precise than indicated by Tyndall and such that
Helmholtz's criticisms seem excessive (the final accepted value of the equivalent
was to be 778).
The fact that Helmholtz strongly criticised in 1847 the only experimental
evidence he knew in favour of his own approach deserves a better explanation
than Tyndall's. In my view three elements influenced Helmholtz: a) the Erhaltung
is a theoretical work, whose origin is largely independent from experimental
results: it was really meant to be a reinterpretation of already existing knowledge,
and thus its value did not have to be connected with "doubtful" experimental
results, i.e. results that did not enjoy universal recognition. Joule, an amateur
scientist, in 1847 still had serious problems in getting scientific recognition and
Helmholtz might not have wanted to rely on such an ally; b) Helmholtz quotes
Joule four times in different passages of the fourth chapter, but despite this it is
possible that he became aware of Joule's papers only in the final preparation of
the Erhaltung : Joule in fact is not quoted in the "Bericht" (written in October
1846). Thus Helmholtz might not have managed to master his results; c) the third
component of my explanation is of a completely different sort. In reworking out
the conversions between the British and the continental units (from Fahrenheit
degrees, feet and pounds to centigrade degrees, kilograms and meters) I found
that Helmholtz made a systematic mistake in the conversions.
164 Joule, James. "On the Changes of Temperature produced by the Rarefaction and
Condensation of Air." In Phil.Mag ser.3 May 1845. Repr. in The Scientific Papers. Vol.1.
Pp.172-189.
165 The two first figures (823, 795) given by Joule in "Equiv Relat" of 1845 at p. 204
of the reprint do not agree with his own values in the papers referred to: "Caloric Effects" of
1843 and "Condens and Raref" of 1845. The value of the magnetoelectric experiments is 838
(see repr. P.156 and 187); the value of the air experiments is 798 (see P.187); the two values
of 823 and 795 correspond instead to the two series of experiments mentioned at Pp.179-180
and P.187 of the reprint of the paper on "Condens and Raref" and discarded from the values
giving the final average of 798. The 1845 value of 774 for narrow tubes was 770 in 1843: see
repr. P.157.

In his search for a mechanical equivalent of heat Joule equated the quantity of
heat needed to increase 1 lb of water by 1 °F (later to be called BTU) with the
corresponding work expressed in feet by force pounds. From his experiments
with the friction of water he obtained a mechanical equivalent of 890 ft lb/BTU
for vessels and 774 ft lb/BTU for narrow tubes.
Considering the conversion factors between British and continental
units (BTU=0.252 kcal; ft lb=O.1382 kgm) to express Joule results in kgm/kcal
we have to multiply them by a factor of 0.5486 (= 0.1382/0.252). Joule's values
of 890 and 774 thus correspond to 488 and 424. The second figure is very close
to 778, the later accepted final value of the mechanical equivalent of heat (in
continental units: 778 x 0.5486= 427 kgm/kcal).
The results of Helmholtz's conversion, as recalled, are different:
instead of the values of 488 and 424 he gives 521 and 452 for vessels and narrow
tubes respectively.
Helmholtz was thus right in asserting that the conversion values of Joule's
experiments given in the Erhaltung were too high, but wrong in asserting that
this was Joule's fault and that Joule's experiments were not accurate. The mistake
was introduced by Helmholtz himself and it is relevant both for his general
evaluation of Joule and for the specific comparison (made below in the same
chapter) of Joule's results with Holtzmann's results (already given in metric
units). In 1881 in a footnote, Helmholtz modified his comments and praised
Joule's work but did not acknowledge his own conversion mistake of 1847 166.
We may wonder why such a careful researcher made a mistake in the conversion.
The mistake is a systematic one: 521/488.3 = 1.067; 452/424.6 = 1.065. My
answer to this question is that Helmholtz was using the French foot 167, a wellknown
unit of measurement, which is equal to 12.8 inches, that is, 0.3251 m. In
fact 12.8/12 = 1.067.
Joule's researches thus did not get the appreciation they deserved in the
Erhaltung ; surprisingly, Helmholtz left the problem unsettled ( he dedicated a
few lines to the whole discussion of the mechanical equivalent) and rapidly
turned to the second question outlined above, the theoretical one, much more
166 Helmholtz WA 1 P.33.
167 On the French foot: Jerrard,H.G. and McNeill, D.B. A Dictionary of Scientific
Units . 5th ed. Chapman and Hall, 1986, P.45.

important in his view: "to what extent can heat correspond to a force
equivalent"? 168
Here force equivalents should not be confused with mechanical equivalents: the
firsts are theoretically identifiable "energy terms"; the second are numerical
conversion factors.
The caloric theory was discussed in explicit reference to the interpretation of
Carnot and Clapeyron 169, for whom the force equivalent is the work produced in
the passage of a certain amount of caloric from a higher to a lower temperature.
Helmholtz criticised 170 W.Henry's and Berthollet's interpretation of the heat
produced by friction as displacement of caloric and asserted that results from the
field of electricity showed that the total amount of the heat of a body can be
actually increased. Helmholtz recalled experimental evidence, entirely based on
research in electricity, against the caloric theory of heat and in favour of the
mechanical one. While frictional and voltaic electricity did not give
unquestionable evidence, because the heat produced could be interpreted as
caloric displaced, "we still have to explain in a purely mechanical way the
production of electrical tensions in two processes, in which any quantity of heat
that can be assumed as transferred never appears" 171 : electrical (not
electromagnetic) induction and movements of magnets. Helmholtz gave the
example of an electrophorus used to charge a Leyden jar for the first, and for the
second the example of electromagnetic machines where "heat can be developed
ad infinitum" 172. It is only at this stage that Helmholtz recalled Joule's experiments
of 1843 173 and asserted that Joule "endeavoured to show directly" that the
electromagnetic current produced heat and not cold even in that part which is
under the actual action of the magnet (no displacement of caloric is thus
conceivable in the electrical circuit). Again the minor role that Joule's results
play in the exposition of Helmholtz's ideas must be noted .
For Helmholtz thus the caloric theory must be rejected, because heat can be
produced indefinitely through mechanical and electrical forces, and the
mechanical theory must take its place. As seen in the previous section, this result
168 Helmholtz Erhaltung P. 27.
169 Helmholtz Erhaltung P.28.
170 Helmholtz Erhaltung P.28
171 Helmholtz Erhaltung P. 29.
172 Helmholtz Erhaltung P.30.
173 Helmholtz Erhaltung P.30 ( Joule's rezsults were wrongly dated 1844)

had already been achieved in the Bericht. What is new and specific of the
Erhaltung is the application of the theoretical framework of tension and living
forces to the mechanical theory of heat. This again is done in a purely conceptual
and qualitative way, without mathematical formulation: free heat will now be
interpreted as the quantity of living force of thermal movement and latent heat as
the quantity of tension forces, namely elastic forces of atoms. Helmholtz quoted
Ampère while making an attempt at clarifying the atomistic view of nature and
the way in which atomic movements can explain radiation and conduction. But
the whole subject is highly speculative and Helmholtz is satisfied with "the
possibility that thermal phenomena be conceived as motions" 174. In fact in this
case the conservation of force "could be verified" whenever conservation of
caloric substance was supposed to hold.
In my view, Helmholtz's cautiousness is completely justified. It is not due to a
lack of conceptual clarity, but to the awareness of the lack of empirical
corroboration. In my opinion, the doubt raised by Helm whether, at this point of
the Erhaltung , Helmholtz was still "uncertain and vacillating" on the validity of
the impossibility of the perpetuum mobile outside mechanics 175 is to be rejected.
Despite being qualitative, Helmholtz's conceptual scheme is wide- ranging. To
explicitly conceive atoms as possessing not only living forces but also tension
forces is a bold step forward and the analogy with free and latent heat seems very
well defined. It allowed in fact an easy reinterpretation of previous knowledge in
the new terms.
The reinterpretation of the heat produced in chemical processes follows: Hesse's
law "partially verified also by experience" had been deduced from the caloric
theory. It asserts that "the heat developed in the production of a chemical
compound is independent of the order and the intermediate steps of the
process" 176. The law, as shown in the "Bericht", is in agreement with the forceequivalent
hypothesis (correlation principle), but here Helmholtz showed that it
can be interpreted in terms of the new concepts of living and tension forces: the
heat produced is now considered a living force, generated by the chemical forces
174 Helmholtz Erhaltung P.31.
175 Helm hinted that Helmholtz's cautiousness in the application of his second root
(central forces and mechanical hypothesis) to thermal phenomena implies that he had doubts
on the first one (impossibility of perpetual motion) as well, and thus on the whole conservation
problem. G.Helm Energetik P.44
176 Helmholtz Erhaltung P.32.

of attraction that play the role of tension forces. Here, implicitly, Helmholtz
applied the mechanical concept of conservation developed in the first chapter: the
vis viva developed between two definite configurations of the system is
independent of the trajectory.
Helmholtz then turned to the final problem of the chapter: the disappearance of
heat. Very little attention had been dedicated to this problem (in correspondence
with the little attention dedicated to the ad nihilum nil fit) :
"As yet nobody has inquired if heat disappears in the production of mechanical
force: which would be a corollary of the law of conservation of force" 177.
In discussing this problem again, Helmholtz revealed his inclination towards a
theoretical approach: both the transformations of work into heat and heat into
work were assumed, but as necessary consequences of the principle and not on
the basis of experimental results.
Helmholtz quoted Joule 178 for the third time, asserting that his results on this topic
were the only ones available and that they seem "sufficiently reliable" 179. The
experience referred to is a famous one: compressed air expanding against air
pressure cools down; this does not take place if the air expands in a vacuum. In
the first case the compressed air has to exert a mechanical force to overcome the
resistance of the air pressure and in the second case it does not. Thus in the first
case the heat which has disappeared can be equated to the work done and thus a
mechanical equivalent can be found, but no equivalent is mentioned here 180.
Finally comes one of the most puzzling sections of the chapter: the discussion of
the research of Clapeyron and Holtzmann. Helmholtz was aware that both
researchers had been working on the assumptions of the caloric theory, in fact in
the "Bericht" he asserted that they dealt with the propagation more than the
production of heat. But here Helmholtz spoke of their research as "tending to find
out the force equivalent of heat" 181 and compared them with his own. Clapeyron's
177 Helmholtz Erhaltung P.33.
178 Joule "Condens and Raref" Phil Mag P.379, rep. P.184.
179 Helmholtz Erhaltung P.33.
180 At Joule's and Helmholtz's ignorance this same approach had been followed,
without actually performing new experiences but reinterpreting in original way old data, by
Mayer in 1842. Mayer had worked out an equivalent of 365 Kgm. If the data Mayer utilised
had been more reliable ( for instance the ones of Regnault) his value would have been correct:
425 Kgm, see Haas Entwickl P.87.
181 Helmholtz Erhaltung P. 33.

approach is discussed at length and criticised: only for gases the law established
by Clapeyron on the assumptions of the caloric theory had received empirical
support, but in the case of gases it is equivalent to Holtzmann's.
Holtzmann assumed that if a certain quantity of heat "enters" in a gas it either
produces an increase of temperature or an expansion. In this second case the
work done gives, following Helmholtz's account of Holtzmann, the possibility to
calculate the mechanical equivalent of heat. On the grounds of Dulong's values
for the specific heats of gases, Holtzmann's equivalent is 374 Kgm 182. Helmholtz
warned that this could be accepted in the framework of the conservation of force
only if all the living force of the heat communicated was actually given as work,
that is, if the sum of the living forces and the tension forces, or, in the old
terminology, the quantity of free and latent heat, of the expanded gas was the
same as the one of the denser gas at the same temperature. This approach is in
agreement with the above quoted one of Joule, and Helmholtz compared the
equivalents of Holtzmann and Joule. The 374 Kgm of the former are compared
with a series of results of the latter, who is credited with having actually
performed the experiments and not to have only reinterpreted previous data. Five
values were given from Joule's results 183: two are the already recalled ones of 452
and 521 (which, as already explained, should be 424 and 488) deriving from the
friction of water, and three (481, 464, 479 which should be 451, 435, 449) are
quoted without reference. My view is that these three values are taken from
Joule's 1845 paper: "On the Existence of an Equivalent Relation". The first
(481/451) refers to the 1843 experiments with an electromagnetic engine, the
second (464/435) to the 1845 experiments on air referred to above, and the third
one to the final average mentioned in Joule's paper 184. As noted above, the
comparison of the results of the two researchers is deeply distorted by the
systematic error in the conversion of Joule's units of measurement. The chapter
ends with a detailed discussion and comparison, which includes a table, of the
laws of Clapeyron and Holtzmann 185.
182 Helmholtz Erhaltung P. 35.
183 Helmholtz Erhaltung P. 36.
184 Going back from Helmholtz's values to British ( and not French) foot-pounds we
get 822, 793, 819 that are reasonably close to Joule's figures of 823, 795, 817 respectively.
185 Helmholtz Erhaltung P.37. Helmholtz was later to claim priority in the
interpretation of Carnot's function: see Wolff "Clausius".

On this point Clausius, who had given 186 in 1850 a mechanical equivalent of 421
Kgm, in 1854 moved a serious objection, asserting that Helmholtz had
misunderstood Holtzmann's law, in which the caloric concept played a role which
cannot be eliminated. This was among the very few criticisms that Helmholtz
accepted during the controversy 187.
In 1882 in an appendix to this chapter, Helmholtz discussed the priority
problem 188.
6 Electricity, galvanism and thermo-electric currents: Helmholtz and the
batteries
The fifth and the longest chapter of the Erhaltung is dedicated to applying the
law of conservation of force to static electricity, galvanism and thermo-electric
currents 189.
Once again Helmholtz displayed an extraordinarily detailed knowledge of
empirical laws of physics in his attempt at further theoretical applications of his
principle.
The first application is an easy one: Coulomb's law, being a strictly central force
law with attractive and repulsive forces, offers the best possible example of how
to formulate a sum of tension forces, to be equated with an increase of vis viva.
But the difficulties do not wait for long: Helmholtz, explicitly referring to Gauss
186 Clausius, Rudolf "Ueber die bewegende Kraft der Wärme und die Gesetze, welche
sich daraus für die Wärmelehre selbst ableiten lassen" Annalen 79 (1850): 368-97, 500-24.
187 Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, pp.76-93. P.90. Still Truesdell believes that
Helmholtz's approach in 1847, while wrong in other aspects, in this was consistent: Truesdell,
Clifford. The Tragicomic History of Thermodynamics. 1822-1854. New York: Springer,1980,
P.162; Truesdell believes that the Erhaltung is "Helmholtz's weakest work" ibid. P.161.
188 Helmholtz WA 1 Pp.71-4.
189 Given that most of the chapter (from P. 48 to P. 58) deals with a careful analysis
of batteries, Kuhn's remark (Kuhn Sim Disc P.73) that Helmholtz did not discuss batteries in
his Erhaltung is difficult to understand.

and unaware of Green's results190, introduced the concept of electric potential. He
defined the quantity
⎛ eieii − ⎝ r ⎠ ⎞
corresponding to the sum of the tension forces consumed, and also of course to
the living forces acquired in the motion of the two charges from an infinite
distance to the distance r, as the potential of the two electrical elements for the
distance r 191.
The principle of conservation can thus be expressed in a new way: "the increase
of vis viva in whichever movement must be considered equal to the difference of
the potential at the end of the trajectory respect to the potential at the
beginning" 192 (the sign of the potential is opposite to that of modern convention).
The potential as defined by Helmholtz is equivalent but for the sign to what later
was to be called potential energy. He showed a good grasp in setting the
relations between potential and work, for instance in the case of the potential of
one body with respect to another193. Here in fact he precisely stated the
equivalence between potential and work. But problems occur in the definition of
the potential of a body on itself (the sum of the potentials of an electric element
of a body with respect to all the other electrical elements of the same body): in
this case Helmholtz's definition is double the modern convention, but what mostly
matters, does not correspond to the work done194 (the potential is supposed to be
double the work done). This shows the "independence" of the two concepts in
Helmholtz's approach. In the original 1847 edition of the Erhaltung there is a
final correction (the only one) referring exactly to these problems. This is a clear
indication of the incertitudes and difficulties faced by Helmholtz with concepts
that in 1847 were by no means common195. Helmholtz in 1882 196 acknowledged
190 According to Koenigsberger, Helmholtz knew of Green's theorem later, in
Koenigsberg, partially through F.Neumann: Koenigsberger H v H P.100.
Erhaltung
191 Helmholtz Erhaltung P.38.
192 Helmholtz Erhaltung P.39.
193 Helmholtz Erhaltung Pp.39-40
194 Helmholtz Erhaltung Pp.42-3. Clausius was the first to criticise this part of the
195 In the 1882 reprint of the Erhaltung in WA 1 the correction is incorporated in the
text (as in the English translations of Tyndall, Kahl and Lindsay), and Helmholtz warned the

in a note that he had considered twice every interaction of two electrical particles
( most contemporary textbooks warn the students not to make such a 'mistake' 197)
and that the use of other authors to equate potential and work was more
appropriate. Still he maintained that his 1847 approach in relating potential and
work was basically correct.
In my view Helmholtz's claim is based on reasonable grounds: I believe he was
among the first to interpret and use "correctly" the "new" mathematical tool (the
potential) in physical research 198.
Helmholtz thus unified explicitly in this essay the tradition of analytical
mechanics (potential function of Gauss, Hamilton and Jacobi) and of engineering
(concept of work), but it is noticeable that, at variance with Clausius' more
mathematical approach of 1852 199, he arrived at the concept of potential not
through the concept of work as a total differential, but straight from the concept
of sum of tension forces. The fact that theoretical rather than mathematical
physics was at the root of Helmholtz's approach is evident from his attempt to
clarify the "mathematical" potential through the introduction of physically sound
concepts and not viceversa. Helmholtz introduced 200 first the "equilibrium
surfaces", later identified with equipotential surfaces, and second the "free
tension", later to be identified by Helmholtz himself with the mathematical
reader with a note to the last appendix WA 1 P.75 (the latter in Kahl's translation is simply
omitted).
196 Helmholtz WA 1 P.75
197 For instance: Feynman, Richard. The Feynman's Lectures on Physics. 3 vols.
Addison Wesley, 1963; vol 2 P.8/15
198 Hoppe's claim that the identification of work and potential started with Riemann is
probably due, being Hoppe a committed Weberian, to the lasting effects of the Weber/
Helmholtz controversy; see Hoppe,E. Histoire de la Physique. Paris: Payot, 1928; P.570.
199 Clausius, Rudolf. "Ueber das mechanische Aequivalent einer elektrischen
Entladung und die dabei stattfindende Erwärmung des Leitungsdrahtes" Pogg Ann 86, 1852:
337-375; transl in:"On the Mechanical Equivalent of an Electric Discharge, and the Heating of
the Conducting Wire which accompanies it." In Tyndall and Francis Scientific Memoirs on
Natural Philosophy 1 (1853): 1-32.
200 Helmholtz Erhaltung Pp.40-1 and P.42

potential function 201. It is worth noting that still in 1847 Helmholtz's idea of
electric tension was reminiscent of Volta's influential "density of electricity" 202.
Helmholtz with great intellectual ingenuity struggled to apply his conceptual
framework of living and tension forces to every realm of nature: he did not use
mathematical requirements as heuristic tools and thus differed from mathematical
physicists such as Clausius and Riemann to a great extent. But he also differed
widely from experimentalists such as Joule: in discussing the heat generated in an
electrical discharge he stressed the interpretation of the heat as the vis viva
produced by a decrease of the quantity of tension forces (i.e. of the potential) and
of the discharge as an oscillatory process of alternating currents, more than the
search for experimental results. In fact for the mechanical equivalent of heat
(whose numerical value enters in the laws expressed, given the electrostatic
system of units utilised 203) he asserted that "up to now observations are lacking".
Detailed information is instead given on the relation between the heat produced
through the discharge of a specific battery and the shape and dimension of the
connecting wire 204 (quoting Riess, Vorselman de Heer and Knochenauer).
The fact that conceptual explanation played a greater role than empirical
corroboration or mathematical formalism is again exemplified in the discussion of
galvanism. For Helmholtz 205 the contact law of Volta, if correctly interpreted, is
not in disagreement, as often remarked 206, with the impossibility of perpetual
motion.
201 Helmholtz, Hermann. "Ueber einige Gesetze der Vertheilung elektrischer Ströme
in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche." In Pogg Ann
89 (1853): 211-33, 352-77. Repr. in WA 1 p.475, P.224.
202 On Volta's influence in Germany see: Teichmann, Jurgen. "L'influenza di A.Volta
in Germania." Quaderni del Giornale di Fisica 3 (1977): 43-60. In 1849 Kirchhoff unified the
concepts of electric tension, electrostatic and electrodynamic potential : Kirchhoff,Gustav. "On
a Deduction of Ohm's Law, in Connection with the Theory of Electrostatics" Phil Mag 3
XXXVII (1850): 463-8; translated from Pogg Ann LXXVIII (1849) P.506; see also : my
Energy and Electr Pp.77 and 96-7; Archibald, Thomas. "Tension and Potential from Ohm to
Kirchhoff." In Centaurus 31 (1988): 141-63.
criticisms.
203 See Planck Princip P.41.
204 Helmholtz Erhaltung Pp.43-4. This was going to be another object of Clausius'
205 Helmholtz Erhaltung P.45.
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":
they do not produce an electrical imbalance but instead originate from an
electrical imbalance. For a correct interpretation of the processes it is necessary
to restrict the contact law to first class conductors (metals) and realise that the
second class conductors conduct only by means of an electrolytic process. Thus a
contact force can be interpreted in terms of attractive and repulsive forces of the
two metals which remove electrical charges of the contact area from one metal
to the other. Equilibrium is reached when an electrical particle in the passage
from one metal to the other does not acquire or lose living force, that is, when
the variation of living force from one metal to the other is compensated by an
identical variation of tension forces independently of the shape and dimension of
the contact surfaces and in agreement with the galvanic series of tensions.
Once more, conservation of force based on central forces gave a conceptual
explicative framework. But the case was different with the long analysis of
galvanic currents.
Conservation of force is applied to batteries not producing polarisation,
producing polarisation but not chemical decomposition and producing both. Here,
as enthusiastically remarked by Helm 207, conservation is applied in the nonmechanical
sense, as equivalence of numerical effects without a reinterpretation
in terms of living and tension forces. The first case, batteries without polarisation,
is the only one for which we have precise laws, experimentally corroborated.
Through the laws of Ohm, Lenz and Joule, Helmholtz gave the amount of heat
that must be generated in the circuit to have conservation of "energy". This heat
has to be equivalent to the chemical heat that would be developed without
electrical effects and the result is that the electromotive forces of the two metals
be proportional to the difference of the heat developed through their oxidation
and through their combination with acids.
The case of polarisation and of polarisation with chemical decomposition is
discussed in detail but only in a qualitative way, i.e. not only without the
application of the conceptual scheme, but even without quantitative indications,
for the lack of reliable empirical data 208.
Joule is quoted again, for his experiments intended to show the equivalence of
chemical and electrical heat 209. But again his results and methods are criticised
207 Helm Energetik P.44.
208 Helmholtz Erhaltung Pp.51-6
209 Joule Phil Mag XIX 1841 P.275; XX 1843 P.204; Helmholtz Erhaltung P.56.

and judged unreliable, despite providing evidence for a part, at least, of
Helmholtz's innovative programme.
Once more Helmholtz turned to his main line of thought: a conceptual
explanation of electrical movements between metals and fluids through attractive
and repulsive forces, in analogy with what he had already achieved in the case of
contact forces. In the case of polarization currents the two metals would attract
positive or negative electrical charges, respectively, till saturation. In the case of
chemical decomposition there is not a stable equilibrium but a continuous
process. The velocity of the process does not continually increase for the loss of
vis viva through development of heat. An equivalence can be derived between the
heat produced (living force) and the consumption of chemical elastic force
(tension force). Thus Helmholtz's conservation of "energy" helped to clarify
another difficult topic.
Finally, Helmholtz discussed thermo-electric currents and the Peltier effect.
Without applying the concepts of tension and living forces he utilized the
principle of conservation to derive two consequences (on the heat produced and
absorbed at equal (constant) temperatures and equal currents) but "I do not
know, so far, experimental measurements for these two consequences" 210.
7 Which force equivalents for magnetism and electromagnetism?
In the last chapter Helmholtz's approach reveals 211 all its fertility and limits and
here the intrinsic difficulties connected with the formulation and application of
the principle of conservation of "energy" can be better understood.
In the section dedicated to magnetism the pattern follows the one for
electrostatics: the inverse square law provides an easy expression for tension
forces. Living forces and potentials are defined, both for two bodies and for a
body on itself. An interesting application is the one for a non-magnetised steel bar
brought close to a magnet, magnetised and separated. There is an expenditure of
-1/2W in mechanical work (again the potential on itself W is twice the work )
that is acquired by the now magnetised bar 212.
210 Helmholtz Erhaltung P.60.
211 Helmholtz Erhaltung P.60-9.
212Helmholtz Erhaltung P.62-3.

The section on electromagnetism starts with a concise and straightforward
assessment of the state of art. It shows Helmholtz's complete mastery of the
subject and outlines the programme of research that he was to pursue for forty
years. His statements are fundamental, then, not only for the history of energy
conservation but also for the history of electromagnetism.
Helmholtz reveals himself to be familiar not only with the celebrated laws of
Ampère but with the more recent ones of Weber, Lenz, F.Neumann and
Grassman. The characterization of the different approaches is very precise:
Weber's law, at variance with Ampère's, explains electromagnetic induction, but
in a conceptual framework at odds with Helmholtz's. Weber in fact assumed
forces depending on velocities and accelerations. Helmholtz remarked that "up to
now " 213 it had not been possible to refer this law to the central force hypothesis.
Both the laws of Neumann and Grassman agree with Weber's law for closed
currents, the only currents for which experiments were available 214.
Helmholtz's plan was thus to confine the application of the principle to closed
currents and to show that the "same laws" could be deduced through the
principle 215. The strategy is clear: lacking a central force law for
electromagnetism, if some "empirical" laws already deduced from non -
Newtonian force hypotheses could be rededuced from the principle of
conservation of "energy", strong evidence would be acquired for the justificatory
power of the principle. Moreover, if new consequences could be successfully
predicted, its heuristic power would gain evidence.
But some not minor difficulties are implied in the process; two question are
relevant: a) is the dichotomy of tension and living forces really needed for
Helmholtz's specific application of the principle to electromagnetic phenomena or
a correlation principle (quantitative equivalence) would be sufficient?; b) does
the principle offer a reliable key to discover all the force equivalents that have to
be taken into account?
213 Helmholtz Erhaltung P.63.
214 This same remark was the starting point of the famous paper of 1870, when
Maxwell's approach too was analysed: Helmholtz, Hermann. "Ueber die
Bewegungsgleichungen der Elektricität für ruhende leitende Körper." In Borchardt's Journ 72
(1870): 57-129; repr. in WA 1 pp.545-628. Helmholtz's suggestion to compare the different
laws in the case of open currents finally led Hertz to devise and perform his famous
experiments in the late eighties.
215 Helmholtz Erhaltung P.64.

The relevance of the first question can be better understood if we remember that Helmholtz's
sharp distinction between a positional and a kinetic energy was the result of the privilege he
had attributed to central Newtonian forces. It was not a result of the other basic hypothesis of
the impossibility of peprpetual motion. Thus alternative expression of the principle of
conservation were possible and were in fact given (correlation, generalised potentials). In
which way then was Helmholtz's approach superior? Were electromagnetic phenomena better
explained if interpreted in terms of the two forms of energy?
The second question does not deal with the superiority of one or the other
expression of the principle but with the possibility of the theoretical principles to
provide new knowledge; in a way it questions the basis of the new theoretical
physics: can the energy terms in a specific situation be provided by a theoretical
analysis based on the principle alone or is experimental knowledge of that
situation needed for a correct application of the principle? In this second case
what is the advantage of using the principle?
An answer to these questions can be obtained through a discussion of two of the
four cases analysed in this section of the Erhaltung.
The first case discussed refers to a system consisting in a magnet moving under
the effect of a current 216. Helmholtz identified the tension forces with the ones
utilised in the current : aAJdt (in mechanical units, with a = mechanical
equivalent of heat, A = the electromotive force of a single cell, J= the current),
that is, (following the result of the section on galvanism in the previous chapter),
with the heat generated in the chemical process inside the battery. The living
force is supposed to be composed of two parts: one, as before, is the heat
generated in the circuit (W= the resistence of the circuit) by the current:
the other is the living force acquired by the magnet under the effect of the
current. This is identified with:
J dV
dt dt
where V is the potential of the magnet towards the conductor carrying a unitary
current (accepting Ampère's view that the electrodynamic effects of a closed
current are equivalent to those of a magnetic double layer). Thus:
A−
J =
1
a dV
dt
W
The term
1
a
dV
dt
216 Helmholtz Erhaltung Pp.64-5.

was interpreted by Helmholtz as a new electromotive force: the force of the
induced current. The law is similar to Neumann's, but more precise : Helmholtz
gave the value 1/a for what in Neumann was an indetermined constant 217.
This "demonstration" became very famous as an indication of the heuristic power
of the principle 218. Instead the fourth case 219 discussed in this section of the
Erhaltung , concerning the interactions between two currents, became famous for
the opposite reason: the lack of correct deductions from the principle.
Here Helmholtz simply extended the previous formulation: tension forces
provided by the batteries of the two circuits are
A 1 J 1 and A 2 J 2,
living forces identified with the heat produced by the current in the two circuits
are
J 2
1 W 1
and J21
W2 plus 1
a J1J dV
2 dt
and 1/a J1J2 dV/dt is interpreted as the living force of a circuit under the effect of
the current circulating in the other one
His claim that the results so obtained were in agreement with Weber's does not
imply that his result was correct; Helmholtz in fact dismissed two kinds of
potentials that, this time, really exist : the mutual potential of the two currents
(electrokinetic energy) and the potential of a current on itself (selfinduction) 220.
The conflicting outcome of the two cases discussed point to a serious problem: to
what extent Helmholtz's principle was a useful heuristic device? In fact the first,
correct, deduction was in a way a reinterpretation of already existing knowledge.
Helmholtz did not include terms that could have been reasonably included not to
violate Neumann's law. What about, for instance, the tension forces given by the
potential JV of the current on the magnet? Have they to be taken in account or
not? From a theoretical point of view we could easily give a formulation of the
217 Helmholtz Erhaltung P.65.
218 In 1873 it was still quoted and praised in Maxwell's Treatise Maxwell, James
Clerk. A Treatise on Electricity and Magnetism. 2 Vols. 1873; Thomson J.J. ed of 3rd edition
1891; repr New York: Dover,1954; Pp.190-3;but J.J.Thomson remarked in a note to third
edition that the law of induction cannot be deduced through the principle of conservation of
energy alone: another equation is needed. Ibidem P.192.
219 Helmholtz Erhaltung Pp.67-8.
220 see Planck Princip P.47

principle taking this potential into account, in analogy with the case (previously
discussed) of the coupling of two magnets. But if we admit that potential we
would not be able to rededuce Neumann's induction law, a law that is empirically
corroborated. 221. In the second case terms unknown in 1847, but that indeed exist,
were not predicted.The usefulness of the specific application of the concepts of
tension and living forces is also questionable 222: what is the rational for defining
aAJdt as tension forces? The equation in fact for Helmholtz still holds if A = 0,
thus without tension forces. Moreover, what is the reason to call the product JdV
"the living force acquired by the magnet"? Helmholtz's efforts show thus that it is
impossible an a priori precise theoretical deduction of the energy equations:
experimental knowledge is required; on the other side, without a guiding
principle it is difficult to find and to interpret this experimental knowledge;
moreover the expression of the principle can offer a ground for comparison for
theories experimentally equivalent.
This basic principle-theory-experiment interplay present in the Erhaltung will be
permanently connected with the subsequent developments of the energy
conservation theory. It is always difficult to take into account the different forms
of energy involved in a specific process, and impossible to find the ultimate
result 223.
At the end of the section Helmholtz asserted that for some more complicated
cases discussed by Neumann and Weber the principle cannot give precise
determinations, but only qualitative indications, also because of the lack of
experimental results 224.
8 Conclusion
Pp.111 and 114.
221 See Planck Princip Pp.45-6.
222 It was in fact discussed in Helm Energetik Pp.45-7.
223 On the impossibility of a "primary" expression of energy see: Planck Princip
224 The electromagnetic equations will be revised and rediscussed in 1854 during the
controversy with Clausius.

The short and concise conclusion of the Erhaltung is mainly dedicated to
physiological problems, Helmholtz's "professional" domain. Here again, together
with a summary and reassessment of results previously expressed in the "Warme"
and in the "Bericht", we find interesting new remarks.
The main problem, as usual, was the formulation of the force equivalents of the
energy balance. Helmholtz asserted that in the case of the vegetable world, any
precise application of the principle is impossible for the lack of the necessary
information. It can only be predicted that the tension forces stored are of
chemical origin and that the only living forces absorbed are those of the
"chemical solar rays" 225.
The application of the principle to the animal world can be made with greater
accuracy. Here Helmholtz, summarising the results of his previous research for
the benefit of physicists, introduced for the first time the concepts of tension and
living forces in physiology: animals utilise a certain quantity of chemical tension
forces and generate heat and mechanical forces. But Helmholtz believed that the
mechanical work done by the animals is only a small quantity compared to the
heat produced and could thus be omitted in the equation of the force equivalents.
The corroboration of the conservation principle was related here to the following
question: whether the combustion and conversion of the nutritive substances
generate a quantity of heat equivalent to that produced by the animals. Helmholtz
did not go into details here 226, but, on the basis of the Dulong and Despretz
experiments, believed he could offer a positive answer, at least approximately.
In my view this approach again shows that Helmholtz's main aim was to outline
a general framework for the largest possible class of phenomena, independently
of the actual experimental determination of the mechanical equivalent of heat and
thus independently of precise experimental corroboration. The exact value of the
mechanical equivalent was not known, but not too relevant in this context: in
asserting that the work done by the animals is a small percentage of the heat
produced Helmholtz showed he was satisfied with a gross figure 227.
Next comes a criticism of Carlo Matteucci's 1847 rebuttal of the conservation
approach. Helmholtz showed that it was based on theoretical misunderstandings
and experimental inaccuracies 228.
225 Helmholtz Erhaltung P.69.
226 Helmholtz Erhaltung P.70
227 Helmholtz Erhaltung P.70. See Kremer "Therm of Life" P.251.
228 Helmholtz Erhaltung P.70-1.

Finally we get to the real conclusions, which are worth quoting at length.
Helmholtz in fact did not claim to have demonstrated the principle, but only
believed he had shown that the principle of conservation was "not in
contradiction with any known fact in natural science but that it is corroborated in
a remarkable way by a great number of such facts" 229.
The sophisticated plan outlined at the beginning of the Einleitung has been
carried out: with the greatest possible completeness the consequences of uniting
the principle with the known laws of natural phenomena have been outlined. But
Helmholtz was aware that his own extraordinary theoretical efforts lacked
experimental corroboration: he did not claim to have achieved the latter, but
instead that " the consequences outlined have still to wait for experimental
confirmation". His goal was "to show to the physicists the theoretical, practical
and heuristic relevance of the principle of conservation of force, whose
exhaustive corroboration must be considered, probably, as one of the main tasks
of physics in the near future" 230.
It would certainly be difficult to find in scientific literature a paper expressing a
clearer consciousness of the goals attempted and the results achieved. Both from
the physical and the methodological point of view, the Erhaltung is a
masterpiece.
Questioning one root: the central force controversy with
Clausius (1852-54)
The third stage of Helmholtz's struggle with 'energy' is characterised by
a strong controversy he had with Clausius. The Erhaltung had not enjoyed great
success. Helmholtz was now a physiologist in Koenigsberg and was trying to
convince F.Neumann of his approach to the conservation principle. In the debate
with Clausius, Helmholtz attempted to defend his own views and at the same
time to qualify as an expert on physico-mathematical grounds, not a minor task
for a physiologist. Clausius' criticisms started in 1852 and were extremely serious
: he disclaimed, among others, not the physical probability but the mathematical
necessity of central forces to fulfil vis viva conservation. In his answer,
Helmholtz showed himself to be a great master of mathematical physics and of
scientific methodology; through the introduction of a few (partly ad hoc) physical
229 Helmholtz Erhaltung P.72.
230 Helmholtz Erhaltung P.72.

hypotheses he managed not to lose ground. But Helmholtz's external success
could only mask what in fact was a very serious internal problem in his research
program. Clausius' criticism was explicitly recognized as sound only thirty years
later, but Helmholtz must have realized its soundness immediately. In fact there is
no further trace of central forces and of the dichotomy of potential and kinetic
energies in the fourth stage of his commitment to the energy problems: the
popularization of the doctrine.
-----------------
The Erhaltung did not raise enthusiastic reactions, apart from amongst the
members of the Physikalische Gesellschaft. Despite Du Bois' pressure, Magnus'
presentation of the work to Poggendorff was very cool. Magnus objected to what
he supposed was an attempt at unifying physics through mathematics.
Poggendorff refused to publish the paper in his Annalen because there were no
new experimental results and it was too long 231. I believe the acceptance would
have been different if Helmholtz had not mistranslated Joule's numerical results
and if he had presented Joule's experiences as corroborating his own views.
The essay, together with the philosophical introduction dropped for the
presentation at the Physikalische Gesellschaft, was published by Reimer in
Berlin, and Helmholtz, to his surprise, received an honorarium 232.
Helmholtz after the Erhaltung returned to physiology and received the chair of
physiology at Koenigsberg in 1849 233. But of course he did not give up his
interest in energy conservation: in Koenigsberg he struggled to convince
F.Neumann, who had already used an extended interpretation of the vis viva
231 Koenigsberger H v H p.38; C.Jungnickel, R.McCormmach, Intellectual Mastery :
vol.1, p.157; on the generational differences among German physicists see: Caneva, K."From
Galvanism to Electrodynamics: The Transformation of German Physics and its Social
Context." HSPS 9 (1978) 63-159.
232 Koenigsberger H v H p.42.
233 Extraordinary Professor and Director of the Physiological Institute: he became
Professor in 1851.

conservation 234, on the theory of "force" conservation 235. Helmholtz worked with
Neumann's help on a paper to appear in the 1851 Annalen 236.
In 1851 Helmholtz travelled around Germany (that is, German speaking states
and towns) and met (together with physiologists) the most important physicists.
In a letter he remarks that W.Weber received him with "less apparent cordiality
than his brother in Leipzig" 237. For certain the feelings between the two were
influenced by the deep contrast, plainly evident to both, between Weber's force
law of 1846 238 and Weber's potential law of 1848 239 with Helmholtz's 1847
approach.
In 1852 Helmholtz added to his own mathematical knowledge, at least in part
through F.Neumann, Green's theorems 240, which were to turn useful in the
controversy that was about to start against Rudolf Clausius. In that very year, in
fact, Clausius made some criticisms to the Erhaltung, criticisms that were based
on potential theory and directed at the core of Helmholtz's program, that is to "the
main advance on the investigations of Robert Mayer" 241.
234 See Helmholtz's references already in the 1847 "Bericht".
235 Koenigsberger H v H p.64.
236 Helmholtz, Hermann. "Ueber die Dauer un den Verlauf der durch
Stromesschwankungen inducirten elektrischen Ströme." In Poggend Ann 83 (1851): 505-40.
See: Jungnickel,C. and McCormmach, R. Intellectual Mastery : vol.1, p. 162.
237 Koenigsberger H v H p. 81.
238 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen über ein allgemeines
Grundgesetz der Elektrischen Wirkung." In Werke 6 vols. Berlin, 1892-4. Vol.3, pp.25-214.
See: Koenigsberger H v H p.100.
239 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen" In Pogg Ann 73
(1848): 193. Tr. in Taylor's Scientific Memoirs 5 (1852): 489-529. W.Weber already in 1848
had shown that his force law of 1846 could be derived, contra Helmholtz's Erhaltung, from a
potential. This implied that the work done by the force was a total differential and that the
force law did not violate the impossibility of perpetual motion. But, given that the force law
included velocities and accelerations, the potential included a kinetic term. Actually the kinetic
and positional part of the potential could not be sharply divided, a serious blow against
Helmholtz's approach. Weber's law soon acquired a leader position in electrodynamics, but his
potential was not accepted as a viable interpretation of energy conservation till the early
seventies.
240 Koenigsberger H v H p.100.
241 Koenigsberger H v H p.118.

This discussion was to outline a third, more mathematical, formulation of the
principle of conservation of energy, after the correlation of forces (conversion
with constant coefficients) and conservation of "force" (variation of vis viva
equals the sum of tension forces).
The problem of appreciating the specific characteristics of Helmholtz's innovative
theoretical approach was a serious one: Helmholtz claimed recognition among the
physicists, both experimental and mathematical. But to receive it he had to fight a
hard battle. Only J.J. Jacobi, the mathematician, encouraged him, recognising the
links between the Erhaltung and the works of the "great mathematicians of the
preceding century", and despite the contrary opinion of Lejeune-Dirichlet and
Eisenstein 242. On top of Magnus' and Poggendorff's criticisms, Karsten in 1850 in
the Fortschritt der Physik im Jahre 1847 inserted the report on the Erhaltung
in the physiological section, and only later, in 1855, commenting on the 1850-51
debate on the "Theorie der Wärme", Helmholtz managed to give news of his
paper in the physical section 243 of that journal.
In this context Clausius' criticisms were received with great preoccupation 244,
given that Clausius was already a well- known mathematical physicist, and
answered in detail.
Rudolf Clausius and Hermann Helmholtz knew each other rather well: together
with Wiedemann they regularly took their meals together in the winter of 1847 245
and since 1848 had been in the habit for a long time of meeting almost daily 246.
Clausius was in fact a participant in Magnus' colloquium from the beginning. In
1850 he published a famous paper on the mechanical theory of heat, in which he
defined the two fundamental principles of thermodynamics. He derived a value
242 Helmholtz 1882 final appendix to the Erhaltung : WA 1 p.71; Koenigsberger H v
H p. 43; for a modern denial of the physico-mathematical relevance of theErhaltung see:
Truesdell, Clifford. The Tragicomic History of Thermodynamics. 1822-1854. New York:
Springer,1980. Pp.161.
243 See also: Heimann "Helmholtz and Kant" p.233 n.104. Helmholtz in 1855 started
publishing a series of six "Berichts" in the Fortschritte der Physik (1855-9) that can now be
seen as the first history of energy conservation.
244 Koenigsberger stresses Helmholtz's preoccupation, so great to make him feel
"betrayed" by a former friend. Koenigsberger H v H p. Actually Clausius' initial criticisms
were very light and mixed with praise.
245 Jungnickel,C. and McCormmach, R. Intellectual Mastery : vol.1, p.256.
246 Koenigsberger H v H p.115.

for the mechanical equivalent of heat and in 1852 in his first paper on
electricity 247, which is at the origin of the controversy, he "applied his heat laws
to electrical discharge and thermoelectricity" 248. He thus united mechanics, heat
and electricity in a thorough study based on mechanical principles. Clausius later
dedicated great efforts to an electrodynamical theory that he saw as part of his
own mechanical theory of heat 249. In fact in 1865 he started republishing his
papers on electrodynamics in the second volume of the first edition of Die
Mechanische Wärmetheorie 250. In 1879 he went much further: for the second
edition of the book he rewrote the entire second volume and dedicated it entirely
to his own interpretation of electrodynamics; thus the book appeared with the
surprising title of Die Mechanische Wärmetheorie Bd2 Elektrodynamik : Die
Mechanische Behandung der Elektricität 251. In this volume, where Clausius
"demonstrated" his electrodynamic force law of 1875 252 based on a kinetic
potential, very little room is left for Maxwell's and Helmholtz's electrical theories.
247 Clausius, Rudolf. "On the Mechanical Equivalent of an Electric Discharge, and the
Heating of the Conducting Wire which accompanies it." In Tyndall and Francis Scientific
Memoirs on Natural Philosophy 1 (1853): 1-32 and 200-9.
activity.
248 Jungnickel,C. and McCormmach, R. Intellectual Mastery vol.1, p.167.
249 Historians have payd very little attention to this not minor aspect of his scientific
250 Clausius, Rudolf. Die Mechanische Wärmetheorie 2 vols Braunschweig, 1865.
The first volume was translated in English in 1867.
251 The first volume had already appeared: Clausius, Rudolf. Die Mechanische
Wärmetheorie 2nd ed 1st vol. Braunschweig, 1876. (An English translation of the second
edition of the first volume appeared in 1879: The Mechanical Theory of Heat London 1879).
The second volume followed three years later: Clausius, Rudolf. Die Mechanische
Wärmetheorie 2nd ed. 2nd vol. Braunschweig,1879. A French translation of both volumes of
the third German edition was made by F.Folie and E.Ronkar, Bruxelles :1897-8.
252 Clausius' law admitted, as Weber's one, forces depending on velocities and
accelerations; but, at variance with Weber, the velocities were 'absolute' (respect to the ether)
and thus the forces were also non central. Clausius' ether explains Helmholtz's 1882 remark
about the loss of intellegibility . See n.94 above. Clausius, Rudolf. "Ueber ein neues
Grundgesetz der Elektrodynamik." In Pogg Ann 156 (1875): 657-60; tr. in Phil Mag 1 (1876):
69-71.

In 1876 Clausius, discussing 253 his own electrodynamic law of the previous year,
clarified his interpretation of the conservation principle, in a way that is strongly
connected with the approach of his early works on electricity: the only condition
needed to fulfil the principle of conservation is that the work done by the forces
is a total differential. No physical interpretation of potential energy or of the
energy concept is given, conservation is still defined along the lines of the
analytical tradition as vis viva conservation. Clausius' approach derives from his
committment to mathematical potential theory 254 and to his identification of work
with the difference of potential.
All this shows that Clausius' 1852 remarks on Helmholtz's results were by no
means casual, but originated and deeply rooted in Clausius' research
programme.
A main difference in the interpretation of the energy concept was to create a
lifelong barrier between these two champions of the mechanical view of nature:
Helmholtz believed in two forms of energy sharply divided and based on central
forces depending only on distance, while Clausius admitted forces that were not
central and depended also on velocities and accelerations as far as they admitted
a potential; the fact that in this (kinetic) potential appeared terms including both
positions and velocities was not seen as a problem by Clausius . The existence
itself of a potential was considered as the fulfilment of the impossibility of
perpetual motion.
The whole controversy of 1852-54 took place in the pages of the Annalen der
Physik 255, a clear indication that both Clausius' and Helmholtz's theoretical
contributions were now fully accepted by Poggendorff 256.
253 Clausius, Rudolf. "Ueber das Verhalten des elektrodynamischen Grundgesetzes
zum Prinzip von der Erhaltung der Energie und über eine noch weitere Vereinfachung des
ersteren." In Pogg Ann 157 (1876):489-94; tr.in Phil Mag s5 1 (1876): 218-21.
254 Clausius published many editions of his textbook on the Potential and the Potential
Function starting from 1859. A French translation of the second German edition of 1866
appeared in 1870: Clausius, Rudolf. De la fonction potentielle et du potentiel. Tr by F.Folie.
Paris:Gauthier-Villars, 1870.
255 Comments on this controversy are in Planck Prinzip Pp.48-50, Koenigsberger H
v H Pp.115-20, Heimann "Helmholtz and Kant" Pp.234-7, Jungnickel,C. and McCormmach,
R. Intellectual Mastery p.163.
256 On the cultural policy of the journal see: Jungnickel,C. and McCormmach, R.
Intellectual Mastery Vol.1, Pp. 113-128.

The controversy started with some remarks on Helmholtz's Erhaltung in the
notes of Clausius' 1852 paper on electric discharge. Helmholtz answered in a
note of his 1853 257 paper on the distribution of electrical currents in material
conductors. In the same year 258 Clausius fought back and in 1854 259 Helmholtz
expressed his views at length , in a paper to be reprinted in the energy section in
the first volume of his collected works. A short comment of Clausius in 1854 260
closed the debate. As already seen 261, later on in 1882, in the appendix to the
reprint of the Erhaltung in the same first volume of the collected works,
Helmholtz admitted, in part and without quoting Clausius, that some of his old
objections were correct.
There were three main points of debate and I will deal with them separately: a)
the definition of potential; b) the conceptual model of central forces; c)
accusations of lack of precision.
a) Helmholtz had introduced the concept of Spannkraft (potential energy) without
going into a discussion of the concept of potential. The peculiarity of Helmholtz's
approach is that he jumps from the vis viva theorem to the conservation of
"force" principle, without following the now standard pattern of f x ds = work,
work as total differential = difference of potential. In the case of central
Newtonian forces work is indeed a total differential and thus the two approaches
intersect considerably. In fact, as discussed in the previous section, Helmholtz
introduces in a straightforward way in the second chapter of the Erhaltung the
sum of tension forces (potential energy) and the energy concept (the variation of
vis viva equals the variation of the sum of tension forces, the sum of vis viva and
tension forces is a constant) as conceptual and physical entities. Instead he
introduces the term potential only in the fifth chapter, with a physical grasp not
completely refined and without knowledge of Green's contributions, but "in
257 Helmholtz, Hermann. "Ueber einige Gesetze der Vertheilung elektrischer Ströme
in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche." In Pogg Ann
89 (1853): 211-33, 352-77. Repr. in WA 1 pp.475-519; n.1 p.488.
258 Clausius, Rudolf."Ueber einige Stellen der Schrift von Helmholtz "uber die
Erhaltung der Kraft." In Pogg Ann 89 (1853): 568-579.
259 Helmholtz, Hermann. "Erwiderung auf die Bemerkungen von Hrn. Clausius". In
Pogg Ann 91 (1854): 241-60; repr. in WA1, 1882, pp.76-93.
260 Clausius, Rudolf. "Ueber einige Stellen der Schrift von Helmholtz "uber die
Erhaltung der Kraft", zweite Notiz." In Pogg Ann 91 (1854): 601-604.
261 See nn.125, 126, 127 above. Helmholtz WA 1 p.70.

conformity with Gauss in his magnetical researches" 262 as the sum of tensions
consumed by the motion from infinity to r, and, equivalently, as the sum of the vis
viva produced. Thus :
"the increase of vis viva in any movement must be considered equal to the
difference of the potential at the end of the trajectory with respect to the potential
at the beginning" 263.
That is, the sum of tension forces is equivalent to the difference of potentials:
− ∫ r
R
φ dr = e1e2 R − e1e2 r
and, of course, also to the gain in vis viva caused by passing from the distance R
to r.
Helmholtz then introduces the concept of the potential of a body in itself and of
the potential of a body on another. In a specific case given as an example, he
calculates the difference of potentials before and after the movement of a
quantity of electricity. This difference is defined as equivalent to the quantity of
work done:
-(V + (Wa +Wb)/2) 264
As already seen Clausius in 1852 criticised Helmholtz assertion that in the above
expression the potential W of a body on itself is not equal to the corresponding
work done, but twice as much (the corresponding work is in fact W/2). The
accusation of not having understood the deep relations between potential and
work was a serious one from Clausius' point of view.
Clausius in the same1852 paper had established a relation between vis viva and
potential different from that of Helmholtz's Erhaltung. He started from the vis
viva theorem and equated the increase of vis viva to the quantity of mechanical
work produced during the same time in the system 265. Clausius did not accept
Helmholtz's "sum of tension forces" (potential energy) and the corresponding
interpretation of the conservation principle. For Clausius, work, being in most
cases a total differential and thus its integral depending only on the initial and
final positions, can be identified with a difference of potentials. The potential of
an exterior system of masses on a given system is introduced as the function:
262 Helmholtz Erhaltung p. 38.
263 Helmholtz Erhaltung p. 38.
264 Helmholtz Erhaltung Pp.42-3.
265 Clausius "Electric Discharge" p.3.

where μ are immovable masses, m the masses of the system and ρ the distance of
the two masses from each other.
In a similar way:
is the potential of the given system of masses upon itself.
"The work consists simply in the increase of these potentials" 266.
Thus in 1852 Clausius explicitly asserted that the potential is work stored in the
system. Work as total differential and difference of potential are identical
concepts. This is a statement of the greatest relevance, but very different from
Helmholtz's. In the Gauss-Clausius tradition energy would never become a
physical quantity. The principle of conservation was often to be called, following
the old tradition, the vis viva conservation 267 and the only really important
requirement was that work be a total differential. This interpretation left open the
possibility that forces other than the central Newtonian ones could satisfy the
conservation principle, if the work done by these forces satisfies the mentioned
requirement. An answer to Clausius was thus due.
In 1853 Helmholtz clarifies his own use of the term "free tension" in the fifth
paragraph of the Erhaltung asserting that its definition "is identical to what
Gauss defined as potential and Green as potential function" 268. Thus in 1853
Helmholtz explicitly acknowledges a result of the unifying power of the
mathematical potential theory: the concept of electrical tension, formerly with
Volta and Ohm density of the elastic fluid called electricity, had been redefined
by Kirchhoff in 1849, both for static electricity and for currents, as difference of
electrical potential 269. Electrostatics and galvanism had thus been unified.
In a note in the same page 270, Helmholtz replied to Clausius' criticisms of 1852
asserting that the problem of the definition of the potential does not imply any
266 Clausius "Electric Discharge" in the note of p.5, quotes both Gauss and Green.
267 See for instance: Riemann Schwere Elektricität Magnetismus ; Sturm, M. Cours
de Mécanique , M.Prohuet ed. Vols. 2. Paris: Mallet-Bachelier, 1861.
268 Helmholtz "Einige Gesetz" p.224.
269 Kirchhoff, Gustav. "On a Deduction of Ohm's Law, in Connection with the
Theory of Electrostatics" Phil Mag s3 37 (1850): 463-8; translated from Pogg Ann 78
(1849) p.506.
270 Helmholtz "Einige Gesetz" p.224.

substantial disagreement : Helmholtz's own definition of potential can be
considered correct and his own development consistent, even if he admits that it
was not a good definition, because it was not in agreement with F.Neumann's.
Helmholtz had simply assumed for the potential of a body (system of bodies) on
itself double the value than had Clausius. That is, he had considered for the
potential of the couple A-B both the work done to transport A to its position
under the force of B and viceversa to transport B under the force of A. But the
work done in the example under discussion had been calculated correctly.
Clausius could not be satisfied with this answer: He spends six pages in 1853 271
clarifying that Helmholtz's approach rests on what from his point of view was a
serious mistake, the conceptual difference between potential and work. For
Clausius the correct result for the work done in the example at issue is only due
to the compensating effect of two opposite mistakes. He first shows that the great
importance of the concept of potential of two masses, each one respect to the
other, is connected with its equivalence "in a specific but frequent case, with that
of mechanical work" 272. Work has to be considered as one of the most important
quantities of the whole of mechanics and mathematical physics, "and also
Helmholtz in his essay utilized it with this meaning" 273.
Thus if we have to introduce the concept of "potential of a mass on itself" with
twice the value of the corresponding mechanical work, priority should be given
to the other concept of potential (that is, to the potential of a body on another
which is equivalent to the work). The opposite choice of Helmholtz is a flaw" 274.
But a more serious mistake, for Clausius, is to have added together in the same
expression a potential corresponding to the work (the potential of a body on
another) and one corresponding to twice the work (the potential of a body on
itself). In the simple example given by Helmholtz the final result is correct only
because the mistake of the double value of the potential of a mass on itself is
compensated by the particular choice of the values of the electricities involved.
Helmholtz's answer in 1854 on this point is interesting: he recalls that in that
particular example his definition was supposed to hold only under specific
hypotheses, that in general his definition of potential and work are correct, for
example for electrified bodies and for magnets, and asserts that in the case of the
271 Clausius "Einige Stellen" Pp.568-74.
272 Clausius "Einige Stellen" p.569.
273 Clausius "Einige Stellen" p.569.
274 Clausius "Einige Stellen" Pp.569-70.

demonstration criticised by Clausius, a wrong work equivalent for the potential
was not presupposed, on the contrary the very goal of the demonstration was to
find the work equivalent 275.
In no way could the different methodologies be better clarified. In a short answer
in 1854 Clausius reasserts his point 276 and finally in 1882 in the last appendix to
the Erhaltung, Helmholtz accepts the criticism that his definition of the potential
of a charge (body) on itself is less convenient than the one that equates it with the
work done. Still he denies giving a wrong value for the work in 1847: it was
asserted in the Erhaltung that the work was half the mentioned potential.
To summarise this point of the controversy: for both Helmholtz and Clausius the
difference of potential has to be related to the work done; Helmholtz in the
Erhaltung faced some difficulties in the definition, partly for his ignorance of
Green's papers (he in fact only quoted Gauss) and partly for his physical rather
than mathematical grasp of the problem. Nonetheless it is remarkable that his
"free tension" was later to be identified with the "potential function" and that, as
everywhere else in the Erhaltung , he started from the theoretical assumption of
the equality between quantity of Spannkraft and variation of vis viva to find the
work done in the specific cases. The equivalence between potential and work is
not, as in Clausius, a prerequisite, but a consequence of the existence of a
potential energy. In the above quoted case of the electric discharge (i.e. where a
variation of the charge distribution exists), Helmholtz knows that the tension
forces would then be modified during the discharge, and thus builds up his
example to avoid the problem., through considerations of symmetry. Without
doubt Clausius was right in pointing out that Helmholtz's definition of potential of
a charge on itself was twice the work done, but in the context of Helmholtz's
research this was a minor mistake and does not imply that Helmholtz had not
understood, despite his lack of knowledge of part of the literature, the relations
between work and potential. Moreover Clausius had an advantage: after
Helmholtz's 1847 essay and before Clausius 1852 paper the relations between
potential and work had been clarified by Weber (1848) and Kirchhoff (1849).
b) The second point of the controversy is a much more serious problem for
Helmholtz: Clausius challenged the necessity of the central force hypothesis, one
of the roots of Helmholtz's definition of energy.
275 Helmholtz "Erwiderung" Pp.76-7.
276 Clausius "Zweite Notiz" Pp.601-2.

Clausius' approach is in fact different: starting from the same work-vis viva
equation as Helmholtz does, he recognizes that in certain cases the work done is
a complete differential. Thus the vis viva can be equated to a difference of
potential. But Clausius does not mention at this stage the potential energy, nor
does he think that it should be a positional term. Even much later, the principle of
conservation of energy for Clausius was meant to be the requirement that the
work be expressed as a total differential. Clausius believed in the kinetic theory
of heat, but he did not apply to it the dichotomy of vis viva and tension forces as
Helmholtz had done in the fourth chapter of the Erhaltung . Clausius asserted in
1852 that
"heat consists in a motion of the ultimate particles of bodies and is a measure of
the vis viva of this motion" 277.
Thus the work done, or the difference of potential between the initial and final
conditions, includes this effect too:
" The sum of all the effects produced by an electric discharge is equal to the
increase of the potential of the entire electricity upon itself" 278.
Mechanical work, electricity and heat are unified here but "potential energy" and
"energy" do not appear. Even farther away from Clausius' perspective is the idea
of a conservation of the sum of sharply separated kinetic and positional terms. In
the paper mentioned, Clausius too relies on the model of central Newtonian
forces depending only on distance, but introduces it only as a case which is
mathematicallly simple, often occurring in physical situations:
"The determination of the work may be much simplified in particular cases which
very often present themselves" 279.
Clausius does not believe central Newtonian forces to be a conceptual model
necessary for the intelligibility of nature. This criticism of Helmholtz's approach
does not appear in the 1852 paper, but after Helmholtz's remarks of 1853,
Clausius in the same year expressed his views on this point rather clearly.
The central issue is immediately faced, "not for personal reasons, but because it
is connected with problems of general scientific interest" 280. The problem is
clearly defined: Helmholtz asserted that the vis viva principle holds only when
the acting forces can be decomposed in forces acting on material points, in the
277 Clausius "Electrical Discharge" p.5
278 Clausius "Electrical Discharge" p.6
279 Clausius "Electrical Discharge" p.4.
280 Clausius "Einige Stellen" p.574.

directions that join them and whose intensity depends only on the distance, that is
only for central forces. But Clausius claims that the demonstration given by
Helmholtz does not justify the assertion.
In fact, during the demonstration, one of the very statements to be derived, that
the intensity of the force is a function of the distance, is presupposed 281. This
presupposition is part of Helmholtz's model as expressed in the Einleitung. Thus
the principle of vis viva is either unnecessary to derive the central forces (if we
already presuppose that the intensity is a function of the distance) or is
insufficient (if we do not make that assumption).
Moreover, Clausius asserted that from the vis viva theorem, assuming that the
vis viva is a function only of the space coordinates, we can derive that the work
too is a total differential and also that the force acting is a function of the space
coordinates. But this does not imply that the force be central. The only possible
implication for Clausius is that, if one of the two conditions hold (direction or
intensity) the other does too.
Finally Clausius, while not questioning the "physical probability", denies the
"mathematical necessity" of central forces 282.
Helmholtz, apart from deriving them from the vis viva principle in the second
chapter, had assumed in the Introduction that a point acts in different directions
with the same force. Clausius does not consider this as evident: the contrary is
not "unthinkable". Moreover in Helmholtz's first chapter of the Erhaltung, when
the principle of virtual velocities is derived from the vis viva principle it is also
asserted that the result is that the forces of two points act on the direction joining
them. But for Clausius this, again, was part of the assumptions and cannot be
considered a deduction. Helmholtz's whole attempt to demonstrate the
mathematical need for central forces is, for Clausius, a flaw.
The core of Helmholtz's program was seriously shaken. No doubt he was deeply
worried by the criticisms for he dedicated a great effort to preparing a twenty
page paper, to be published in 1854 and reprinted in 1882 in the energy section
of the WA 1, after the Erhaltung.
Replying to Clausius in this paper, Helmholtz asserted that he could derive from
the vis viva principle the need for the forces to be central, without further
hypotheses, because he expressed the principle only with reference to the relative
positions of the points and not to the absolute ones. In Helmholtz's view this
281 Clausius "Einige Stellen" p.575.
282 Clausius "Einige Stellen" p.577.

means taking into account the actual bodies and not "purely imaginary systems of
coordinates" 283. Helmholtz admits that he assumed that the vis viva depends only
on the positions of the points and thus only on their distance. If this version of the
principle is accepted Helmholtz can derive, through a massive use of
mathematics, both the hypotheses leading to central forces (the intensity depends
on the distance and the direction is in the joining line) also for the case in which
there is a point moving under the action of a material element. In this case, in
principle, the vis viva of the point should depend on the orientation of the
distance. Furthermore, Helmholtz explicitly asserts that the only implicit
principle used, the superposition of the effects for the force, if accepted
reinforces his conclusions 284.
Clausius quick answer is very short, he denied that in the Erhaltung the concern
was only with relative positions and reasserted his views on the impossibility of a
mathematical deduction of the central forces 285.
In 1882 once more Helmholtz had to accept the criticism, but with some
qualifications: he now admits that the demonstration in the second chapter needs
a restriction, that is: forces that depend also on velocities and accelerations are in
agreement with the vis viva theorem, but not with the principle of action and
reaction 286. Nevertheless they are in agreement with the impossibility of perpetual
motion. Helmholtz asserts in appendix 2 that this point was clarified for him by
Lipschitz. He does not quote the controversy with Clausius, and in appendix 3
while quoting Clausius' electrodynamic law does not quote Weber's.
In a section below I will analyse the role of Helmholtz's expression of the energy
concept and of the conservation principle in the electromagnetic debate of the
70's. It is sufficient for the moment to point out that the root of the problem, the
supposed necessity for energy conservation of central forces, was already explicit
after Weber's 1848 expression of a kinetic potential and Clausius' 1853
criticisms.
c) Other minor points were part of the controversy: Clausius in 1852 287, while
praising Helmholtz's Erhaltung, claims that some parts were incorrect; he
283 Helmholtz "Erwiderung" p.82.
284 Helmholtz "Erwiderung" Pp.88-9.
285 Clausius "Zweite Notiz" p.604.
286 Helmholtz WA 1. Appendix 2 and 3 to the Erhaltung; pp.68-70.
287 Clausius "Electrical Discharge" p.6

quotes 288 and criticises a proposition of Vorsselman de Heer accepted in the
Erhaltung ("the total heat which is provoked in the entire circuit by an electric
discharge is independent of the nature of the circuit"). The same for an expression
of Riess 289 referring to the independence of the heat of discharge from the nature
of the connecting wires.
Clausius in 1852 also criticises Helmholtz's interpretation of Holtzmann,
asserting that the latter did not believe in the consumption of heat, but in its
invariability. This is a point which Helmholtz seems to have misunderstood and
that seems to imply a serious flaw in his analysis of the work equivalent of heat in
the fourth chapter of the Erhaltung. Clausius ends his 1853 paper with an
acknowledgement and an appreciation of Helmholtz's achievements in the
Erhaltung, despite the criticisms 290.
Helmholtz in 1854 answers, in a detailed and balanced way, the specific points
about de Heer and Riess and admits his own mistake as far as the interpretation
of Holtzmann is concerned 291. He finally adds four new points to the
electromagnetic research of the Erhaltung , on the basis of Poisson's magnetic
induction, of his own researches of 1853 on the oscillations of induced currents.
He found that a galvanic current has an electrodynamic potential in itself,
proportional to the square of the current intensity. If the circuit is interrupted
there is a conversion of this "force equivalent" in heat either in the spark or in the
extracurrent 292. Finally he shows that F.Neumann's law of induction through
magnets or currents is in agreement with conservation of energy.
Clausius' final remarks acknowledge the clarifications and only insists on the
problem of central forces 293.
The result of the controversy is remarkable: if not the great success claimed by
Koenigsberger 294, Helmholtz achieved some success. He had shown himself to be
also a skilled mathematical physicist, able to debate with a researcher as clever
as Clausius. In the circumstances this was a particularly difficult achievement for
Helmholtz: Clausius was right (as later indirectly acknowledged by Helmholtz
288 Clausius "Electrical Discharge" p.16
289 Clausius "Electrical Discharge" p.21
290 Clausius "Einige Stellen" Pp.578-9.
291 Helmholtz "Erwiderung" p.90.
292 Thus correcting his "wrong" deductions of 1847.
293 Clausius "Zweite Notiz" p. 604.
294 Koenigsberger H v H p.116.

himself) in the specific instances of a) the derivation of the central forces from the
principle of vis viva and b) of the definition of potential .
Helmholtz's strength was founded on his broad, general viewpoint and on the
great heuristic and justificatory power of his approach, which allowed a
systematic application to a variety of natural phenomena, and is not founded on
the individual, specific demonstrations and applications. This was, with great
chivalry, acknowledged by Clausius himself. The debate with Clausius being
based on specific instances, Helmholtz was in a weak position ( Clausius was
actually formally right in most criticisms) and it is remarkable that he managed
not to let Clausius prevail. His target of being accepted among mathematical
physicists was achieved. The 1853 paper was the third he published in the
Annalen, and as already recalled, the whole controversy was published in that
journal, which occasioned a greater diffusion in Germany of the Erhaltung 's
ideas.
Helmholtz's fame spread perhaps in Britain before at home. W.Thomson read the
Erhaltung in 1852 295 and at DuBois-Reymond's suggestion Tyndall translated
and published it in 1853, together with Clausius' 1852 paper. Helmholtz met
Tyndall in 1853 in Berlin, which began a lifelong friendship, and when, in the
same year, he participated in the meeting of the British Association for the
Advancement of Science at Halle he discovered he was better known than in
Germany 296.
But the result of the controversy with Clausius was not without a lesson for
Helmholtz, as will be seen in the next section.
Popular conservation of "force" : where have the central
forces gone ? (1854-64)
In a series of lectures, most of which are not included in the
Wissenschaftliche Abhandlungen , Helmholtz presented to a large public the
universal validity of the theory of conservation of energy. In so doing, he only
utilized his very first approach to the problem: the conversion of forces through
constant coefficients, the related extension of the impossibility of perpetual
295 Planck Prinzip p.68.
296 Koenigsberger H v H p.109 (Tyndall), p.113 and 120.

motion to all natural forces, the kinetic theory of heat and its mechanical
equivalent. That is, he only utilized one of the two conceptual roots of the
Erhaltung, and did not talk of the requirement that all the forces be central and
that the energies be reduced to only two kinds. Moreover, he abandoned the
"narrow laboratory" of the physicist and threw himself at cosmological heights.
He did not give examples of applications of the conservation principle to the
physical and chemical laws as he had done in the Erhaltung. Instead he gave a
large-scale, unified picture of the world in energy terms: from the Kant-Laplace
cosmogonic hypothesis to thermal death, to an analysis of the planetary system,
of the fall of meteors, of winds, tides and all sorts of technical machines and
devices. At variance with what had happened in 1847, these lectures were
received enthusiastically both by the scientific milieu and by the general public.
The first lecture was soon translated by Tyndall and published in the
Philosophical Magazine and this was useful for spreading Helmholtz's fame
across the Channel. Did this new-old approach have to be chosen, after Clausius'
criticisms, to please both the scientists and the public ? Would it not have been
easier to popularize energy referring to its only two asserted forms and to only
one model for the forces?
But a second shift takes place at the beginning of the seventies:
empirical rather than theoretical aspects are stressed in the enunciation of the
principle. Was this second shift connected with a more empiricist methodological
view adopted by Helmholtz in the late sixties? What was Helmholtz's relation to
the other scientists who had been and still were working along similar, but not at
all identical lines? Some remarks on these questions are related to the discussions
of Mayer's priority which Helmholtz kept introducing into his popular lectures.
We are thus led to the priority debate, where some of these problems receive an
answer.
_______________________
On February the 7th 1854 in Königsberg, Helmholtz gave a popular
lecture on the interaction of natural forces 297 that was soon to become extremely
famous. On the first of June of the same year, in fact, Helmholtz told his father
that a second edition of the lecture had already been requested 298. The lecture
297 Helmholtz, Hermann: Ueber die Wechselwirkung der Naturkräfte un die darauf
bezüglichen neuesten Ermittelungen der Physik. Königsberg: Gräfe & Unzer,1854.
298 Koenigsberger H v H p.124

was soon translated into English by Tyndall and appeared in the Philosophical
Magazine 299, was reprinted with slight modifications 300 in the first (1871) and
second (1876) edition of the second volume of the Populären Wissenschaftlichen
Vorträge, appeared, with a note on "Robert Mayer's Priorität" 301, in the first
(1884), second (1896) and third edition (1903) of the Vorträge und Reden 302.
Helmholtz's renewed interest in the conservation of energy was
without doubt due to the controversy with Clausius just discussed 303. But the
result is surprising : to the demand for some more popular accounts "of the great
principle that was to underlie the science of the future" 304 Helmholtz answers with
a magnificent exposition, ranging to cosmological dimensions, of the law of the
correlations of forces and not of his own mechanistic view of conservation of
energy based on central forces and on the dichotomy between kinetic and
potential energies! Despite the skillfulness of his counter-attack on Clausius it
seems that Clausius' objections had a great impact on our author. In fact two
other popular lectures on the same argument (in 1861 and 1862-64) 305 share the
same approach.
The argument runs as follows, and it is worthwhile quoting it at
length:
"The new philosopher's stone of the seventeenth and eighteenth
centuries is perpetual motion. Perpetual motion was to produce work out of
nothing, but a point was reached where it could be proved that at least by the use
of pure mechanical forces no perpetual motion could be generated. The idea of
work became identical with that of the expenditure of force. How, then, can we
299 "On the Interaction of Natural Forces and recent Physical Discoveries bearing on
the same" Phil Mag s.4 75 Suppl.Vol.11 : pp.489-518; quotations here from Tyndall's
translation.
300 See the author's preface in 1871
301 To be discussed below.
302 The English translation was also included in the first edition (1873), and in the first
volume of the second (1881) and third (New Edition 1893) editions of the Popular Lectures
on Scientific Subjects translated by Atkinson. A French translation appeared in 1869, together
with the translation of the Erhaltung (Paris: V.Masson et Fils); more recently an Italian
translation in: Helmholtz, Hermann. Opere . V.Cappelletti ed. Torino: UTET, 1967.
303 Koenigsberger H v H p.120.
304 Ibidem.
305 See below.

measure this expenditure and compare it in the case of different machines? In the
case of a watermill with an iron hammer, the work must be measured by the
product of the weight into the space through which it ascends. The work
performed by the hammer is determined by its velocity. The motion of a mass
regarded as taking the place of working force is called the living force (vis viva)
of the mass. Living force can generate the same amount of work as that expended
in its production. It is therefore equivalent to this quantity of work. Mathematical
theory has corroborated this for all purely mechanical, that is to say, for moving
forces. After this law had been established by the great mathematicians of the last
century, a perpetual motion, which should make use of pure mechanical forces,
such as gravity, elasticity, pressure of liquid and gases, could only be sought after
by bewildered and ill-instructed people." 306
But here comes the new problem caused by the conversion processes
in the 19th century:
"But there are still other natural forces which are not reckoned among
the purely moving forces, heat, electricity, magnetism, light, chemical forces, all
of which stand in manifold relation to mechanical processes. Here the question of
a perpetual motion remained open." 307
At this stage Helmholtz's argument shows the complete abandonment
of one of the two conceptual roots and main assumptions of his Erhaltung,
namely the hypothesis of central forces depending only on distances. In fact the
conservation of force is here seen as the correlation of forces through constant
coefficients, based on the acceptance of the impossibility of perpetual motion :
" ..it was asked, if a perpetual motion be impossible, what are the
relations which must subsist between natural forces? Everything was gained by
this inversion of the question. It was found that all known relations of forces
harmonize with the consequences of that assumption, and a series of unknown
relations were discovered at the same time, the correctness of which remained to
be proved." 308
Contributors to this line of thought were Carnot in 1824 (despite the
incorrect view of the nature of heat), Mayer in 1842 309, Colding in 1843 310, Joule.
306 Helmholtz "Interaction" pp.489-95
307 Helmholtz "Interaction" Pp.495-6.
308 Helmholtz "Interaction" p.498
309 This is the first published appreciation of Mayer's priority. Helmholtz will repeat it
in 1855 (in the "Bericht" see n.2); in1861 ("On the Application of the Law of the Conservation

It seems as though the importance of the different approaches disappeared for
Helmholtz:
"...several heads .... generated exactly the same series of reflections" 311.
Helmholtz asserts he played only the following role:
"I myself, without being acquainted with either Mayer or Colding, and
having first made acquaintance with Joule's experiments at the end of my
investigation 312, followed the same path. I endeavoured to ascertain all the
relations between the different natural processes, which followed from our
regarding them from the above point of view." 313
From this analysis Helmholtz's specific interpretation of energy
conservation completely disappears. This cannot be attributed to a simplification
due to the attempt at popularizing a difficult subject. The reduction of all the
interactions of natural "forces" to only two kinds of energy and to one only, well
known, model of force would have simplified the task. This approach is, in my
opinion, motivated by the desire to offer to the public a picture of the new theory
which was not controversial for the specialists; it is the first implicit
acknowledgement of a weakness in Helmholtz's position of 1847 and the
beginning of a retreat to safer ground.
The subsequent steps of the argument recall: the recognition that work,
apart from not being created, cannot be destroyed; the mechanical theory of heat;
and the success in determining the work-heat equivalent. Again Helmholtz's own
results are expressed in simplified terms :
of Force to Organic Nature." In Proc Roy Inst 3(1861): 347-57; rep. inW A 3, pp.565-80, at
p.573); in 1862-4 (see: "On the Conservation of Force." In Pop Lect 1873, p.320); in a letter
to Tait, published in Tait's Sketch of Thermodynamics in1868; in 1882 in an appendix to the
reprint of the Erhaltung : WA 1 Pp.71-4. But in 1883, after the controversy with Dühring, its
judgement on Mayer's contribution was much less appreciative. See below.
310 On Colding see: Dahl, Per. "Ludwig A. Colding and the Conservation of Energy."
In Centaurus 8 (1963): 174-88.
311 Helmholtz "Interaction" p.499
312 I am inclined to think that this is true in view of the lack of reference to Joule in
the 1847 "Bericht" and of the little room dedicated to Joule in the fourth chapter of the
Erhaltung compared with Clapeyron and Holtzmann, surely less relevant to the problem of
the heat-work equivalent.
313 Helmholtz "Interaction" p.499

" ...the quantity of force in nature is just as eternal and inalterable as
the quantity of matter. Expressed in this form, I have named the general law 'The
Principle of Conservation of Force'." 314.
Clausius is clearly and explicitly praised for the first expression of the
second principle, a principle that does not contradict the law of conservation of
force :
"Only when heat passes from a warmer to a colder body, and even then
only partially, can it be converted into mechanical work" 315.
After giving credit to Carnot and W.Thomson, to the latter also for the
problem of thermal death, Helmholtz abandons mathematical-mechanical
developments and instead of giving "a glance at the narrow laboratory of the
physicist" gives " a glance at the wide heaven above us, the clouds, the rivers, the
woods and the living beings around us" 316.
An extraordinary series of applications of the great law now takes
place, ranging from a reassessment of the Kant-Laplace hypothesis (with a
calculation of the heat produced by the assumed condensation of the bodies of
our system from scattered nebulous matter) to the evaluation of the heat produced
by the speed of the meteors, and to the comparison of the heat coming to the
surface of the earth from within, with that reaching the earth from the sun. The
conservation of force is then applied to organic bodies : we can calculate from
the mass of the consumed nutritive material how much heat, or its equivalent
work, is generated in an animal body; the influence of the sun explains why the
combination of the animal and vegetable organic realms does not produce
perpetual motion. There is thus a specific sense in which we can all consider
ourselves to be "as the great monarch of China, sons of the sun"! 317 The ebb and
flow of tides are explained too, through the combined action of the sun and the
moon. The motions of the tides, as already done by Mayer 318, are connected to
the law of conservation of force: they "produce friction, all friction destroys vis
viva, and the loss in this case can only affect the vis viva of the planetary
system." Finally the thermal death of the sun is discussed.
314 Helmholtz "Interaction" p.501
315 Helmholtz "Interaction" p.502
316 Helmholtz "Interaction" p.503
317 Helmholtz "Interaction" p.511
318 Helmholtz "Interaction" p.512

From the refusal of perpetual motion we have been "conducted to a
universal law of nature, which radiates light into the distant nights of the
beginning and of the end of the history of the universe." 319.
Without doubt Helmholtz proved to be a populariser of the first rank
and a master not only of physics, physiology and cosmology but also of the
German language 320. In the fight against the antiscientific philosophical approach
still present in German universities this talk was to play not a minor role 321. The
capacity of giving a scientific, unified picture of the world in relatively simple
terms and with the application of only one law was doubtless of great appeal to
the German mind 322.
Helm correctly underlines the stress given in this talk to the ideas of
Grove and Mayer on the conversion and correlation of forces. This was to be
done by Helmholtz for decades, with his ever-growing authority, "also without
dealing with his own ideas of 1847" 323.
Helmholtz's effort, along the lines just mentioned, was in fact carried
on in other "popular" talks in the following years. Of interest for us is the one
given on the 12th of April 1861 at the Royal Institution in London 324: "On the
Application of the law of the Conservation of Force to Organic Nature" 325
For the second time 326 we find here acknowledged, but not without
reservations, the use of the term energy:
"It might be better perhaps to call it, with Mr.Rankine, 'the
conservation of energy', because it does not relate to that which we call
commonly intensity of force; it does not mean that the intensity of the natural
319 Helmholtz "Interaction" p.516.
320 Helm, who is not always sympathetic with Helmholtz, as already seen, speaks of
"papers and talks stylistically and pedagogically perfect, even classics". Helm Energetik part
1cap 9 par 7.
321 Koenigsberger H v H p.124.
322 To the British too, judging from the rapidity of the English translation of Tyndall
and from the success Helmholtz always had in Britain.
323 Helm Energetik p.
324 This had to be given without any preparation for the insistence of Faraday,
according to Koenigsberger H v H p.199.
325 See n. 3O9.
326 The first being in 1856 : Helmholtz, Hermann. "Bericht über “die Theorie der
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
gained by any natural process, and by which a certain amount of work can be
done." 327
The introduction of the term energy does not imply the use of the
dichotomy kinetic/potential energy. As in the previous talk, Helmholtz does not
use his central forces approach and its implications, but the conversion of forces
through constant coefficients, with the work as unity of measurement of the
different effects:
"But while, by every alteration in nature, that force which has been the
cause of this alteration is exhausted, there is always another force which gains as
much power of producing new alterations in nature as the first had lost" 328.
Examples are given of different forms of motive powers: a raised
weight, velocity, elasticity of a bent spring, elasticity of air and of compressed
gases, heat, chemical forces. After a short summary of the cosmological problems
Helmholtz goes back to the priority issue and acknowledges the works of Grove,
Joule, Mayer, to which "the first exposition of the general principle" 329 is due.
A remark on the "close connection between both the fundamental
questions of engineering and the fundamental questions of physiology with the
conservation of force" introduces the main theme of the talk, the application of
the law to organic nature. We can change a certain amount of food into carbonic
acid, water, and nitrogen, either by burning the whole in an open fire, or by
giving it to living animals as food and burning afterwards only the urea, and in
both cases we get the same result. Thus the total equivalent in mechanical work
spent by an animal, that is, the sum of the animal heat and the actual mechanical
work, must be equivalent to the heat produced by the burning of the food in the
open fire. For animals at rest Dulong and Despretz found that the two quantities
are nearly identical, but some experimental difficulties still exist 330. When work is
performed by animals, for instance ascending a hill, the consumption is five times
greater than when they rest. But only one fifth of this consumption is really spent
in mechanical work, the other four fifths result in the production of heat. An
interesting analogy with a thermodynamic engine is drawn: only one eighth of the
equivalent of the chemical force is really converted into mechanical work, the
327 Helmholtz "Organic Nature" p.563.
328 Helmholtz "Organic Nature" p.566.
329 Helmholtz "Organic Nature" p.572.
330 A problem lasting from the 1846 and the 1847 papers.

other seven eighths are lost in the form of heat: "the human body is a better
machine than the steam engine, only its fuel is more expensive than the fuel of
steam engines" 331. But despite the fact that we cannot yet prove that the work
produced by living bodies is a total equivalent of the chemical forces which have
been set into action, still "I think we may consider it as extremely probable that
the law of the conservation of force holds good for living bodies" 332. Finally the
"vital principle" is disregarded, in that, if there is complete conservation of force,
the physical forces in the living body cannot be removed and again set in action
at any moment by the influence of the vital principle. In fact conservation of force
can exist only in those systems in which the forces in action (like all the forces of
inorganic nature) have always the same intensity and direction if the
circumstances under which they act are the same.
Indeed Helmholtz can claim that in a few years the principle has
completely modified the view of life phenomena, and that a general unified
approach to physical, cosmological and physiological phenomena has been
achieved.
In November 1862 Helmholtz became Pro-Rector of Heidelberg
University and on this occasion delivered a talk on "The Relation of Natural
Sciences to Science in General" 333. After analysing the differences of the Kantian
epistemological approach and that of Schelling and Hegel, Helmholtz recalls and
expands some of the ideas already expressed in the Introduction to the 1847
Erhaltung. Natural science cannot be satisfied with a collection of facts: it
requires the law that rules them and the corresponding causes. There is a
difference between artistic and logical induction : free will implies the
impossibility of reducing our psychological expressions to a rigid law. One of the
deep differences between natural sciences and human sciences is exactly this :
the former can attain quite general rules and laws , the latter judge on the basis
of a psychological sensibility. Human sciences cannot unify observations and
331 Helmholtz "Organic Nature" p.
332 Ibidem
333 Helmholtz, Hermann. "Ueber das Verhältniss der Naturwissenschaften zur
Gesammtheit der Wissenschaften". Rectoratsrede. Heidelberger Universitätsprogramm 1862.
Rep. in Populäre Wissenschaftliche Vorträge 1st vol. Braunschweig: Vieweg, 1865 (and
following editions); translated in English as "On the relation of the physical sciences to science
in general" in the Annual Report of the Smithsonian Institution for the year 1871, p.217-234;
reproduced in PL 1873 (and following editions); also in R.Kahl SW.

experiences in general laws of unlimited validity and extraordinarily great
extension. There is no doubt that looking at nature means to look at a series of
rigorous causal connections, without exceptions. The effort to find a causal
connection everywhere, or to assume it, is one of the features and lessons of
natural sciences.
These remarks on causal laws are a leitmotiv of Helmholtz's analysis of
energy conservation. He will in fact discuss the topic in 1862, 1864 and later
again.
Again the unifying power of reason is at the root of the next series of
lectures, whose introductory talk still bears a famous title: "Ueber die Erhaltung
der Kraft" 334.
Mentioned in a letter to W.Thomson on the 14th of December 1862 335,
they were given in Karlsruhe in the winter of 1862-63 and again, as mentioned in
a letter to Ludwig of the 27th of February 1864 336, "in London at Easter in
English" to an audience of "three hundred, and among them a number of scientific
men" 337.
Helmholtz's point of departure in the talk is completely philosophical,
with a strong inclination towards neo-Kantianism. Helmholtz first refers to the
element of distinction between natural and mental sciences as outlined in the
previous work 338: the special conformity with laws of natural phenomena. It has
334 Helmholtz, Hermann. "Ueber die Erhaltung der Kraft." In Populäre
Wissenschaftliche Vorträge 2nd vol. Braunschweig: Vieweg, 1871. A summary of the English
talks: "Lectures on the Conservation of Energy. Delivered at the Royal Institution." was
published in the Medical Times and Gazette vol 1, 1864; a German version appeared with a
preface in the 1871 Populäre Wissenschaftliche Vorträge (and following editions), of which
an English translation by E.Atkinson: "On the Conservation of Force"was included in 1873
Popular Lectures on Scientific Subjects (and following editions). Quotations here are from
this translation. In the preface to the 1884 Vorträge und Reden we are told that of this series
the only other existing revised talk is: "Ueber die Entstehung des Planetensystems" in
Populäre Wissenschaftliche Vorträge 1876, Vol 3 (and following editions),translated in
Popular Lectures on Scientific Subjects 1881 (and following edition).
p.225.
335 Koenigsberger H v H p.214.
336 Koenigsberger H v H p.220.
337 Helmholtz to Du Bois-Reymond, May 15 1864, quoted in Koenigsberger H v H
338 Helmholtz, Hermann. "On the relation" see n.333.

in fact been possible to discover laws that allow prediction of the origin and
progress of many extended series of phenomena with the greatest accuracy. It is
in this conformity with law that the intellectual fascination which links the
physicist to his subject is based. As in mental science so in natural science, a
single fact can provoke curiosity or astonishment, but intellectual satisfaction can
only be given by the conformity with law, by the connection of the whole.
The Kantian trend is more and more evident : which is the innate
faculty of thought that allows us to discover laws and to apply them? "Reason",
of course. And the best arena for the forces of "pure reason " is the inquiry into
nature.
In a very interesting passage Helmholtz makes an assertion of the
greatest importance as to the motivations of a scientist: the reward is not only a
successful activity or the acquisition of power over a sometimes hostile world,
but also the artistic satisfaction of surveying Nature as a regularly ordered
whole, an "image of the logical thought of our own mind" 339.
This regulative power of reason, this highest scientific activity, this
ideal of Kantian and neo-Kantian philosophy has now produced a concrete
example that is beneath our eyes : an all-embracing regulative principle, the Law
of the Conservation of Force. This law is specially suited to give us an idea of
the specific character of natural sciences. It is of utmost interest then that
Helmholtz's enunciation of the law is greatly different from the original one:
"(it) asserts that the quantity of force which can be brought into action
in the whole of Nature is unchangeable and can neither be increased nor
diminished" 340.
To explain the meaning of quantity of force, Helmholtz refers to its
technical application, the amount of work, and analyses the subject in a way that
recalls the previous popular lectures, but with greater technical (not
mathematical) details and a generous use of good drawings and schemes of
contemporary machines.
Helmholtz's appreciation of the work of Mayer and Joule in this lecture
is relevant, also for a comparison with a later assessment 341. Robert Mayer is here
credited with the first statement of the possibility of a universal application of the
law of the conservation of force, and Joule with having made:
339 Thus there is something more than"the intellectual mastery of nature"!
340 Helmholtz PL 1873 p.320.
341 "Robert Mayer's Priorität" of 1884, see below.

"a series of important and difficult experiments on the relation of heat
to mechanical force, which supplied the chief points in which the comparison of
the new theory with experience was still wanting" 342.
Joule's experiments are actually given great attention, in contrast with
the 1847 Erhaltung. This time, of course, the conversion factors are correct. For
the determination of the equivalent through the production of heat from work
utilising friction, three series of experiments are recalled: water in a brass vessel
(equivalent=424.9 gramme-meters necessary to raise of one degree °C one
gramme of water), mercury in an iron vessel (425 and 426.3), conical rings
rubbing against each other surrounded by mercury (426.7 and 425.6). The reverse
determination, that is, the conversion of heat into work through the expansion of
perfect gases, is also discussed; results (also due to Regnault's measurements)
give : with air 426.0; with oxygen 425.7; with nitrogen 431.3; with hydrogen
425.3.
The agreement between the two set of experiments, realised with such
different methods is really remarkable. There can be few doubts that "heat is a
new form in which a quantity of work can appear" 343.
A comment on the presentation of the principle is necessary here: the
expression of the principle above 344 shows once again that the stress is on the
correlation of forces, but Helmholtz goes even further in hiding his own
conceptual model of central forces that had played such a great role in the
original Erhaltung . In the extension of the law to all natural processes, the
question of the nature of heat became specially important. "In the answer lay the
chief difference between the older and newer views in these respects". Moreover
in virtue of the relevance of this question:
"Many physicists designate that view of Nature corresponding to the
law of conservation of force with the name of the Mechanical Theory of Heat" 345.
Thus Helmholtz stresses an element, the conceptual model of heat,
which despite being universally acceptable and also accepted, was by no means
his own specific contribution to the problem. After an acknowledgement of the
results of the kinetic theory of gases due to Krönig, Clausius and Maxwell 346 and
342 Helmholtz PL 1873 p.320.
343 Helmholtz PL 1873 p.349.
344 See note 340.
345 Helmholtz PL 1873 p.342.
346 Helmholtz PL 1873 p.350.

a discussion of the production of work through chemical and electrical forces, the
paper ends with the relations between the principle of conservation and perpetual
motion. The abandonment of one of the roots of the 1847 Erhaltung is further in
evidence:
"the possibility of a perpetual motion was first finally negated by the
law of the conservation of force, and this law might also be expressed in the
practical form that no perpetual motion is possible, that force cannot be produced
from nothing" 347.
We can conclude that a main intellectual shift happened after the
controversy with Clausius regarding the interpretation of energy conservation:
Helmholtz decided to hide in the popular lectures of the 1854-64 period the
central force requirement and the dichotomy potential/kinetic energy, to hide also
the theoretical applications of the principle specific to physics (the narrow
laboratory) and instead to insist on the cosmological implications and
technological applications, that is, on the explanation of the way in which
machines work.
But the seventies introduce a second even more relevant
methodological and conceptual shift, already outlined by DuBois-Reymond in his
Commemorative Lecture on Helmholtz 348. He remarked on a modification of
methodological approach between the first and second part of Helmholtz's life:
"just as the principle of the conservation of energy has been a safe clue to
Helmholtz's train of thought in the preceding period, so in the later part we have a
similar guide. The fundamental principle of these researches is the empiricist
attitude, which Helmholtz favours in preference to the nativistic, which he
rejected. This is the same contrast that obtained in the sixteenth century between
Leibniz's pre-established harmony and Locke's sensualism, and to which Kant
gave a decided turn in favour of the former doctrine".
This change of methodology is probably linked with the deep new
scientific achievements reached by Helmholtz through tireless activity in the late
sixties. Relevant here are the third part of the Handbuch der Physiologischen
Optik 349of 1867, the papers "On the Facts that underlie Geometry" 350 and " On
1867.
347 Helmholtz PL 1873 p.
348 Koenigsberger H v H pp.237-8
349 Helmholtz, Hermann. Handbuch der Physiologischen Optik 3 Leipzig: Voss,

the Discontinuous Movements of Fluids" 351of 1868, and the long paper
comparing the different electrical theories of 1870 352. The new scientific results
implied among others a reassessment of some aspects of Helmholtz's Kantianism.
I will confine myself here to discussing the implications of this new trend for
Helmholtz's evaluation of energy conservation.
In September 1869, in Innsbruck, at the Opening of the Natural
Science Congress, Helmholtz gave the address "The Aim and Progress of the
Natural Sciences" 353 which once again deals with energy conservation. But it is in
1871, in the Preface to the second volume of the Populäre Wissenschaftliche
Vorträge, that it is possible to find the first influences of a new approach to the
energy debates. Helmholtz in fact, recently appointed to the Berlin chair of
physics, introduces the first printed version of the second Erhaltung 354 with a
remark on Leibniz and Kant. He shows a first detachment from Kant's positions
in the assertion that Kant had misunderstood conservation issues: " the
conference develops a portion of the one on the Interaction of Natural Forces.
This portion deals with the fundamental ideas of the subject, still misunderstood
also by people learned in mathematical mechanics. This is not surprising, because
"even such a mind as that of Kant found difficulty in comprehending them, as is
shown by his controversy with Leibniz" 355
The influences of Helmholtz's methodological shift on the evaluation of
energy debates can be seen in the different judgements that he gave of Mayer's
accomplishments in the popular talks of 1854-64 and in two later short works.
These are: the notes to the 1847 Erhaltung written in 1881 and published in
1882 in the first volume of the Wissenschaftliche Abhandlungen and : "Robert
350 Helmholtz, Hermann. "Ueber die Thatsachen, die der Geometrie zu Grunde
liegen." In Nach der k Ges d Wiss zu Gött 9 (1868): 193-221; repr. in WA 2 pp.618-
351 Helmholtz, Hermann. "Ueber discontinuirliche Flüssigkeitsbewegungen." In
Berliner Monatsberichte (1868): 215-28; rep. in W A 1 pp.146- .
352 Helmholtz, Hermann. "Ueber die Bewegungsgleichungen der Elektricität für
ruhende leitende Körper." In Borchardt's Journ 72 (1870): 57-129; repr. in WA 1 pp.545-628.
353 Helmholtz, Hermann. "Ueber das Ziel und die Fortschritte der Naturwissenschaft."
In Populäre Wissenschaftliche Vorträge 2nd vol. Braunschweig: Vieweg, 1871. Tr. in PL
1881.
354 Originally presented in 1862-3 and 1864, see above.
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,
written in 1883 and published in the 1884 edition of the Vorträge und Reden.
As already recalled, in1881 Helmholtz moves away from some aspects
of the Kantian approach : he asserts that the philosophical introduction of the
1847 Erhaltung was influenced by Kant to a higher degree than "I would think
legitimate now" 356 .
Helmholtz also acknowledges Mayer's merits 357, and explains that he
did not know of his works in 1847, but that from 1854 on he had always given
him due credit. He also quotes at length a letter he wrote to Tait 358 to defend
Mayer's priority. In 1868 Helmholtz still stresses Mayer's theoretical
contributions:
".. the glory of the discovery remains with those who have found the
new idea; the experimental proof is, afterwards, a much more mechanical task".
We cannot ask that "the discoverer of the idea be obliged to do also
the second part of the job". Otherwise we should reject also most of the works of
mathematical physicists, including some of Tait's great friend W.Thomson.
The 1868 letter follows the approach of the popular lectures discussed,
in giving due credit to the theoretical aspects of the conservation principle. But
the situation in 1881 is radically different : a bitter controversy had opposed
Helmholtz to Zoellner 359, an astronomer with spiritualist inclinations and
defendent of Weber's electrodynamic law, and to Dühring, a historian of
philosophy who had written in 1873 a critical history of the principles of
mechanics 360. The main point of this second controversy was about scientific
356 First appendix to the Erhaltung in WA 1 p.68. Both Kahl's and Lindsay's
translations radically differ on this point : for them Helmholtz "still" considers Kant's strong
influence as correct.
357 Fifth appendix to the Erhaltung in WA 1 p.
358 Tait reproduced it in the Introduction to his Sketch of Thermodynamics.
Edinburgh, 1868. This is part of a very long and bitter controversy on the priority of the
"discovery" of the conservation principle, mostly on the pages of the Philosophical Magazine
between 1862 and 1865. See below.
359 See n.371 below.
360 Dühring for many years claimed that the "guild of professionals" led by Helmholtz
persecuted Mayer. Dühring was also the target of a polemic book of F. Engels. In the end he
lost his teaching position in Berlin University. See: Lindsay, Robert Bruce. Julius Robert
Mayer, Prophet of Energy. Oxford and New York: Pergamon Press,1973; pp.12-16.

methodology. Dühring claimed the character "a priori" of the conservation law
and was one of those who recognised in Mayer "a hero in the field of pure
thought" 361.
Helmholtz after these controversies shows a second shift from his own
positions of 1847: now not only the conceptual explanation of central forces is
given up, but also the theoretical character of the conservation principle: the
empirical component acquires greater and greater importance.
In fact the judgement on Mayer is radically different. Mayer's papers'
weakness is exactly what is praised by metaphysicians:
"the illusory demonstration, metaphysically formulated, of the a priori
necessity of this law".
The law 's success was due to Joule's results. Only then was attention
paid to Mayer's work. The origins of the law, even those of the theoretical
demonstration, are inductive: they came from the empirical acknowledgement of
the impossibility of perpetual motion. The (1775) statement of the Paris Academy
was based on such a probable inductive "conviction" (and not on a
demonstration). The conviction was largely shared. Helmholtz himself, he recalls,
since his school years, had heard discussions 362 on the problems of proving
perpetual motion. His target in writing the Erhaltung was not to propose an
original idea but to carry out a critical work, and he was thus surprised that only
Jacobi, "the mathematician", received it well.
Later, in 1883, while asserting, against Dühring, that he had been the
first in 1854 to recognize Mayer's priority, Helmholtz again attacks the
metaphysicians, who believe the conservation law to be an a priori knowledge.
This is the real point of the debate, not Mayer's priority: the problem is the old
fight between "speculation and empiricism", deduction and induction. In this
light, any special concern for Mayer's personal difficulties should be overcome
and the history of the law of conservation of force clearly stated. The
impossibility of perpetual motion had already been stated in the last century for
"conservative forces" after the works of Leibniz and Daniel Bernoulli. Helmholtz
agrees explicitly with Mayer's identification of the two philosophical roots of the
conservation principle: the "ex nihilo nil fit " and the "nil fieri ad nihilum". He
asserts that the first was thus already accepted in the previous century, and that
the second, equivalent to the assertion that work could not be destroyed, despite
361 Helmholtz WA 1 p.
362 See also Koenigsberger H v H p.8 and 25-6.

not being explicitly formulated, was already in the background (of the vis viva
conservation) for conservative forces: in this case work cannot be destroyed. The
problem of the nature of heat helped, through many difficulties, to gain wide
acceptance for the second philosophical root. At the beginning of the forties the
existence of the caloric was widely questioned, not only in scientific debates:
Helmholtz himself recalls an essay on the topic that he had to write when a
student at the Potsdam Gymnasium. Apparently here Helmholtz forgets his own
difficulties in giving up with the caloric in his first papers. He again defines his
Erhaltung not as an original research but as a critical survey, aimed at clarifying
and reordering known facts and deciding between different explanations.
Mayer was not, then, completely original even if his paper of 1842
deserves due credit. But the credit is for the priority in the assertion of an idea,
not for his demonstration. Without an empirical demonstration of the
indestructibility of forces, the " deduction" from obscure metaphysical principles
as "causa aequat effectum" is worthless. Mayer did not give this empirical
evidence in 1842 (he did not explain but only asserted that the work equivalent
was 365 kgm). But even if he had explained his calculation of the work
equivalent he would not have shown anything: "it was necessary to show that
different processes give the same value". The accomplishment of this great
experimental task is the lasting merit of Joule.
Helmholtz becomes more and more severe: only after Joule's work do
Mayer's views acquire the character of "non improbable hypotheses" 363 and
Mayer's destiny teaches young researchers that the best ideas risk to be sterile
without convincing demonstrations.
The priority debate was actually a debate on scientific methodology, as
almost explicitly asserted by Helmholtz. Both in the undervaluation of his own
contribution of 1847 and of Mayer's results Helmholtz clearly shows that in the
early eighties the preoccupations with an empiricist viewpoint were very strong.
Was this due to the bitter controversy that even led him into a depressive state 364
or are its roots to be found in real scientific problems resulting from his previous
theoretical commitments? Helmholtz's original approach to energy conservation
had in fact been seriously challenged in the electrodynamics debates of the
seventies.
363 Helmholtz
364 Koenigsberger H v H p.306.

The first history of the conservation principle: "evolution and
development, not discovery" (1862-65)
In the pages of the Philosophical Magazine for the years 1862-65
there is a bitter polemic between Tyndall on one side and Tait (specially) and
W.Thomson (mostly in the background) on the other 365. Other participants were
Joule, Colding, Verdet, Bohn, Rankine, Akin. The summary of this controversy
can well be considered one of the first contributions to the historiography of the
principle of conservation of energy. With various qualifications, agreements and
disagreements, with the papers' titles ranging from History of Conservation of
Energy, to History of Mechanical Theory of Heat, of Dynamical Theory of Heat,
of Thermodynamics, of Energetics, of Force it can be asserted that at the time
there was awareness of the following contributions to the principle, which "was
not discovered but evolved and developed":
a) conservation of vis viva: Descartes, Huygens, Leibniz, John
Bernoulli, James Bernoulli, Daniel Bernoulli, D'Alembert, Fresnel, L.Carnot
b) dynamical theory of heat : Bacon, Locke, Rumford, Davy, Young
c) correlation of forces : Rumford, Haldat, Morosi, Seguin, Placidus
Henrich, Mohr, Faraday, Liebig
d) mechanical equivalent : S.Carnot, Clapeyron, Holtzmann, Mayer,
Colding, Joule
e) generalization of the principle : Helmholtz, Clausius, Rankine,
W.Thomson.
What was the assessment of Helmholtz's work in this debate? He did
not receive special attention: the focus of the debate was a comparison of the
relative merits of Mayer, defended by Tyndall, and Joule defended by Tait and by
365 See also: Lloyd, J. "Background to the Joule-Mayer Controversy." In Notes and
Records of the R.S. 25 (1970): 211-25.

himself. Helmholtz is seen rather as a source of reliable opinions: his (wrong)
1847 judgement that Joule's works of 1843-5 were not completely reliable is
recalled; he is credited with being the first who generalized the principle to a
number of applications; it is recognised that in 1854 he was the first to give due
credit to Mayer and the first to indicate Mayer, Colding and Joule as pioneers . In
this debate no precise evaluation of Helmholtz's work is made, nor is he involved
in priority controversies with the British, as happened to Mayer and Clausius.
This is perhaps due to his good relations with both contenders.
___________________________
As already mentioned, Helmholtz met Tyndall in Berlin in 1853 and,
thanks to the latter's translation of the Erhaltung , already in 1853, during his first
trip to Britain, he realized he was better known there than at home. Helmholtz's
fame grew after Tyndall's translation of Helmholtz's popular talk of 1854 "On the
Interaction of Natural Forces". An indication is the success, at the time of the
controversy, of the two popular talks on the subject of energy conservation that
Helmholtz gave in London, at the Royal Institution, in 1861 and 1864. The talks
had been organised by Faraday and Tyndall respectively.
Helmholtz and Tyndall kept interacting:in 1870 Helmholtz edited the
German translation of Tyndall's Faraday as a Discoverer (to which he added an
introduction 366), in 1871 Heat as a Mode of Motion , in 1874 the Lectures on
Sound , in 1874 the Fragments of Science (with the preface:"On the Attempt to
Popularize Science" 367).
But Helmholtz since his second trip 368 to Britain, in 1855, had become
a good friend also of W.Thomson, whom he met often. For instance in 1864,
366 Helmholtz, Hermann. Vorrede zur deutschen Übersetzung von J.Tyndall Faraday
as a discoverer. Braunschweig: Vieweg, 1870; pp.V-XI.
367 Helmholtz, Hermann. Vorrede und Kritische Beilage zur deutschen Uebersetzung
von J.Tyndall Fragments of Science. Braunschweig: Vieweg, 1874; pp.V-XXV and 581-97.
Repr. in V u R 1884. Tr in: Nature 10 (1874): 299-302.
368 Helmholtz was a good traveller; among his trips: in 1840 and 1842 in Germany; in
1851 in Germany and Italy; in 1853, 1855, 1861, 1864, 1871, 1884 to Britain; in 1866 and
1881 to Paris; in 1878, 1881 and 1883 to Italy; in 1880 to Spain; in 1881 to Austria, in 1893
the dramatic one to Chicago; plus a number of holidays in Pontresina, Switzerland, a meeting
point for other scientists, such as the Italian Blaserna.

while in Britain for the above talks at the Royal Institution, he managed to visit
the Thomsons in Glasgow 369.
Helmholtz also edited the German translation of Thomson and Tait's
Textbook of Theoretical Physics . The first part of the first volume appeared with
an introduction in 1871 370. The second part in 1874 was accompanied by
Helmholtz's answer 371 to Zoellner's attacks (partly dealing with the debate on
energy conservation). In 1885 Helmholtz's review of the first two volumes of
W.Thomson "Mathematical and Physical Papers" appeared in Nature 372. It
included an interesting account of the Carnot-Joule-W.Thomson-Clausius ideas
on the dynamical theory of heat and on the dissipation of energy.
Less friendly was Helmholtz's relationship with Tait. In 1868 Tait
published in his Sketch of Thermodynamics a critical letter of Helmholtz's
underlining Mayer's merits in the formulation of energy conservation 373. In the
same book he credited Helmholtz with being "one of the most successful of the
early promoters of the science of energy on legitimate principles" 374, but at the
same time he criticized the central force hypothesis: "...is not in the present state
of experimental science more than a very improbable hypothesis" 375. Helmholtz is
instead credited with successful applications of the principle 376 . Tait's judgement
is somehow modified in 1876 in the second edition of the Lectures on some
recent advances in Physical Science, where the two assumptions of Helmholtz ,
now "one of the foremost of living mathematicians and natural philosophers" 377,
369 And also to visit Oxford and Stokes at Cambridge.
370 Helmholtz, Hermann. Vorrede zum ersten Theil des ersten Bandes der deutschen
Uebersetzung von: W.Thomson und P.G.Tait Treatise on Natural Philosophy. Braunschweig:
Vieweg, 1871; pp.X-XII.
371 Helmholtz, Hermann: Kritisches. Vorrede zum zweiten Theile des ersten Bandes
von W.Thomson and P.G.Tait Treatise on Natural Philosophy Braunschweig: Vieweg, 1874;
pp.V-XIV. Tr. in Nature
372 Helmholtz, Hermann. "Report on Sir William Thomson's Mathematical and
Physical Papers". Vol.1 and 2. In Nature 32 (1885): 25-8.
achievements.
373 See previous section for Helmholtz's changing evaluation of Mayer's
374 Tait Sketch p.
375 Tait Sketch par.96 p.56.
376 Tait Sketch par.104 p. and par.127 p..
377 Tait Advances p.67.

are considered equivalent. Still, the separation of energy into two main forms,
potential and kinetic, is attributed to Rankine and W.Thomson, following a
British nationalist historiographic tradition. In any case Tait's stress is on the
empirical elements of Helmholtz's approach: even the assumption of the
impossibility of perpetual motion for the whole of natural forces is judged an
experimental result.
This insistence on the historiographical revaluation of the experimental
elements in the various formulations of the principle is shared in the early eighties
by Helmholtz himself, as seen in the previous section. This is, in my view, owing
to some concrete difficulties faced by Helmholtz in applying his 1847 formulation
of the principle to actual physical research in the late sixties and seventies.
A new tool : potential theory (1852-72)
Helmholtz's Erhaltung did not withstand the criticisms pointed out by
Clausius on the definition of potential and on the privilege accorded to central
Newtonian forces. Its electrodynamic part was also "for the most part
defective" 378 for its lack of terms referring to energy stored with the circuits. The
very derivation of Neumann's law of induction, for which Helmholtz and
W.Thomson had received so much credit (not the least from Tait in 68-69 and
Maxwell in 1873), was not correct 379.
Helmholtz realized this before his critics. He acknowledged the
weakness of the electrodynamic aspects of the Erhaltung during the controversy
with Clausius and defended his own theoretical approach with the introduction
of some new energy terms, but from that moment on his research in the field of
energy shifted a great deal. He was in fact more inclined towards the physicomathematical
problems connected with Green's theorem and with the
mathematical theory of potential as heuristic tools, rather than with the
theoretical applications in the 1847 style. Helmholtz worked hard to master
potential theory, and in the end achieved an extraordinary success: he was able to
378 Whittaker, E. A History of the Theories of Aether and Electricity 2nd ed. New
York: Harper, 1960; p.218.
379 See nn.217-221.

apply it to acoustics (1859), hydrodynamics (1859 and 1868) and
electrodynamics (1861, but mainly from 1869-1870 on) in works that became
masterpieces in their fields.
_______________________
In the German-speaking world the mathematical potential (potential
function) and the physical potential (potential) spread rapidly and were widely
used. Clausius had a prominent role in this, through the different editions of his
treatise on "Potential and Potential function", but already F.Neumann in 1845,
Weber and Kirchhoff in 1848 and Riemann had made various use of these
concepts. C.Neumann too followed this line of thought; in the sixties and
seventies delayed potentials and generalised potentials were also introduced.
Given the now generally applied identification of potential with the work done, in
this tradition all that was required to satisfy energy conservation was to show that
the force in question (the problems were mainly connected with the attempt to
formulate a general electrodynamic force) admitted a potential, that is that the
work done by the force was a total differential. This in fact is equivalent to
admitting the impossibility of perpetual motion.
In Britain the situation was somewhat different. Despite the early and
important contributions to potential theory by Green, Hamilton, W.Thomson and
Stokes, priority was in general given to the "classic" concept of energy, implying
the clear distinction between potential and kinetic parts. Of these two the kinetic
was to be seen as basic : J.J.Thomson among others attempted to show this
supposed priority, while Maxwell in 1873 had defined the idea of a delayed
potential as "inconceivable".
An energy battlefield for the electrodynamic debate (1870-75)
In Electrodynamics Helmholtz opposed the views of Weber, who since
1846 had given a force law depending not only on distances but also on velocities
and accelerations, and of Clausius who in 1875-6 published a contribution along
similar lines. Various physicists contributed to various aspects of the debate, but
here we will only refer to the role of the energy principle. Not a minor one
indeed, but rather the main battleground. Given the mathematical equivalence of
the competing theories and the difficulties in experimenting with open currents,
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
methodological shift of great interest in scientific debates. But the interest is
augmented by the fact that it implied the formulation of more sophisticated
versions of the principle itself. The admissibility of Weber's law had been
questioned on the grounds that his force law did not match Helmholtz's criterion
of central forces. Accepting the supposed equivalence of 1847 between central
Newtonian forces and the impossibility of perpetual motion, the conclusion was
drawn that Weber's law clashed with this basic assumption. Remarks of this kind
were still put forward in the sixties in Britain by Maxwell (1864), W.Thomson
and Tait (1867), Tait (1868). The above mentioned fact that Weber's force
admitted a potential was unnoticed, or, given that this potential included both
kinetic and positional terms, was still considered to violate the requirement of a
sharp distinction between kinetic and potential energy. Finally both Helmholtz
(1870 and 1872) and Maxwell (1873) agreed that Weber's law was not in
contradiction with the impossibility of perpetual motion. But Helmholtz, who
himself had proposed a very general potential law for electrodynamics, kept
criticizing Weber on energy grounds: in Weber's theory energy could become
infinite (1870), kinetic energy could become negative (1872) and so forth till
1882. Weber defended himself vigorously in 1871, reformulating the principle of
conservation of energy and explicitly stating he did not necessarily have to fulfil
Helmholtz's version. He received Carl Neumann's (1871, 1875, 1877) reasonable
support and Zöllner's unreasonable one ( 1876). On another side Clausius very
plainly states that the only condition to fulfil energy conservation is that the work
produced by the force be a total differential (1876) and not that forces be central
or that kinetic and positional terms of energy be sharply split. It is to be noted
that Clausius' target was not so much Helmholtz, but Weber, an indication of the
latter's important role in electrodynamics. Helmholtz's electrodynamic potential
law, based on action at a distance and on the polarization of an interposed
dielectric, in the seventies began to be seen as an understandable translation of
Maxwell's difficult and sometimes contradictory approach. An example of the
difficulties of Maxwell's Treatise (1873) are the many different concepts of
energy presented: as potential energy (electrostatic) and kinetic (electromagnetic)
or as the product of an intensity by a quantity factor; electrostatic energy seen as
charge by scalar potential or as the square of the electric intensity;
electromagnetic energy seen as current by vector potential or as the square of the
380 See my Energy and Electr

magnetic intensity. Maxwell realized the need for a specific interpretation of the
energy concept suited to a contiguous action theory and based on the continuity
equation. Helmholtz translated Maxwell's approach in his action at a distance
with polarized dielectric approach but was never able, even in 1894, to grasp
Poynting's new concept of localized conservation (1884-5) or Hertz's efforts at
"purifying" Maxwell's theory from action at a distance influences (1890-2). The
interpretation of potential energy in Helmholtz's theory was and still is a
challenge. J.J.Thomson in 1885 had based his comparison of electrodynamic
theories on energy grounds and Planck had been very quick at assessing in 1887
the whole problem of energy conservation, both historically and logically.
Helmholtz could not have failed to notice these developments, but at this stage he
was shifting towards a different research programme.
______________________________
In the seventies Helmholtz returned to a problem he had first faced in his 1847
paper: the application of the principle of energy conservation (PCE) to
electrodynamics. From 1847 to 1870 radical changes occurred both in the field of
Classical Electromagnetic Theory (CET) and in PCE. Helmholtz’s purpose in
1870 was to reorder the field of CET 381, with a clarification of the debate on the
basis of considerations referring to conservation of energy. This attempt lasted
for at least twenty years and was in the end successful: Helmholtz in fact played a
major role in comparing the different theories, suggested a set of experiments that
were to become extremely famous and gave an initial impulse to Lorentz's
researches.
The complexity of the CET debate at the beginning of the seventies was later
described by Helmholtz himself:
"This plentiful crop of hypotheses had become very unmanageable, and in dealing
with them it was necessary to go through complicated calculations, resolutions of
forces into their components in various directions, and so on. So at that time the
domain of electromagnetics had become a pathless wilderness. Observed facts
and deductions from exceedingly doubtful theories were inextricably mixed up
together. With the object of clearing up this confusion I had set myself the task of
381 Helmholtz,Hermann. Vorwort zu: Heinrich Hertz, Prinzipien der Mechanik.
Leipzig: Barth,1894; pp. VII-XXVII. Tr. in Hertz, H. The Principles of Mechanics,
Presented in a New Form. Tr. by D.E.Jones and T.Walley. London: 1899. Repr. New York:
Dover,1956. P.4.

surveying the region of electromagnetics, and working out the distinctive
consequences of the various theories, in order, wherever that was possible, to
decide between them by suitable experiments." 382.
In fact, in 1870, he started publishing a series of papers that constituted a
comprehensive study of electrodynamics. "With the object of clearing up this
confusion", he gave a theory, whose mathematical version included Weber, F.
Neumann and Maxwell’s theory as limiting cases. Helmholtz’s starting point is
relevant: it is the assumption of the existence not only of a force between current
elements but of a force and a torque. Both these assumptions contrast with
Ampére’s theory. Ampére admitted a potential only for closed currents and only
a central force between current elements. Helmholtz’s starting point was a
generalisation of F. Neumann’s theory: he gave the most general expression for
the energy of two current elements, consistent with the condition that the force
between two closed circuits should be the one given by Ampere:
where A is a constant depending on the unit of current, r the distance between the
elements of circuit Ds and Dσ traversed by the currents i and j. K too is a
constant, for K=+1 Helmholtz’s potential reduces to F. Neumann’s; for K=-1,
Weber’s potential is obtained; for K=0 plus a dielectric medium, Maxwell’s
theory can be obtained 383. An expression giving both F. Neumann’s and Weber’s
electrodynamic energies is:
where h is an arbitrary constant. The first term is Neumann’s value. The second
term is the difference between Weber’s and Neumann’s values. The second term
vanishes for closed currents, the only case analysed by F. Neumann 384.
In the fourth paragraph Helmholtz analyses the different theories with respect to
energy values 385. The whole energy Φ is the sum of the electrodynamic
382 Ibidem
383 Helmholtz, Hermann. "Ueber die Bewegungsgleichungen der Elektricität für
ruhende leitende Körper." In Borchardt's Journ 72 (1870): 57-129; repr. in WA 1 pp.545-628.
Pp. 549 and 567.
384 Ibidem pp.565-6.
385 Ibidem pp.579-85.

and the electrostatic
In modern notation 386:
In the absence of electromotive forces external to the system:
is essentially negative. In this case, if K is negative Φ might be negative and thus
a perturbation of the system might lead away from equilibrium. That is, the
equilibrium is unstable for K

solve the difficulty of the elastic-solid theory of reflection and refraction of light.
This footnote was the starting point of H. Lorentz’s 1875 doctoral thesis. A clear
schematization of the physical model of Helmholtz’s theory was given by Hertz
in 1892 388. Hertz distinguished two limiting cases, both refer to a conception of
action-at-a-distance starting from the charges on the plates that polarize the
dielectric between the plates. In the first case, the energy is supposed to be
concentrated on the plates and the actions (at a distance) produced by the
polarised dielectric are comparatively small. Ideally, removing B leaves some
action-at-adistance. In the second case, the energy is supposed to be concentrated
in the dielectric particles and so the dielectric action is the prevailing one. This
second case is mathematically equivalent to Maxwell’s. Helmholtz himself was
quite clear about the physical differences between his and Maxwell’s theory:
"The two theories are opposed to each other in a certain sense, since according to
the theory of magnetic induction originating with Poisson, which can be carried
through in a fully corresponding way for the theory of dielectric polarization of
insulators, the action-at-a-distance is diminished by the polarization, while
according to Maxwell’s theory on the other hand the action-at-a-distance is
exactly replaced by the polarization" 389
and
"It follows from these investigations that the remarkable analogy between the
motion of electricity in a dielectric and that of the light aether does not depend on
the particular form of Maxwell’s hypothesis, but results also in a basically similar
fashion if we maintain the older viewpoint about electrical action-at-adistance"
390.
Helmholtz’s comparison between the three theories had already led to
remarkable results: he judged Weber’s theory untenable (for the instability of
the equilibrium given by K < 0) and instead showed great interest in Maxwell’s
limit of his own theory. In addition, he considered this last approach suitable to
solve the difficulty of optical theory.
For my purposes, it is relevant to notice Helmholtz's shift of criticism towards
Weber’s law. Weber’s law is no longer considered inconsistent with the principle
388 Hertz, H. Untersuchungen über die Ausbreitung der elektrischen Kraft. Leipzig,
1892. Tr. by D.E.Jones Electric Waves. London: Mac Millan, 1892. Repr. New York: Dover,
1962. Pp.23-5.
389 Helmholtz "Beweg Elektr" pp.556-7.
390 Helmholtz "Beweg Elektr" p.558.

of conservation of energy: it is recognised that the work done during a complete
cyclical operation is zero, in other words the existence of a potential is
recognized. What is criticized now is the possibility of deducing absurd
consequences from the law. In 1872, Helmholtz modified his PCE with the
explicit requirement that T > 0 (T = kinetic energy), in order to theoretically
disregard the supposed inconsistencies derivable from Weber’s law 391.
What was Weber answer? As mentioned in previous sections, in 1846 Weber
published his non-positional force law and in 1848 he published a derivation of
this law from the expression of a potential. Both these results contradicted
Helmholtz’s 1847 formulation of PCE. In Weber’s case the forces were not
central and there was no clear distinction between kinetic and potential energy.
So the problem was: had Weber’s law to be considered in agreement with a
general PCE, despite its contrast with Helmholtz's PCE? Still in 1867 Thomson
and Tait gave a negative answer to this question:
"... the conclusion (of Weber’s theory) would stand in contradiction with the
’Conservation of Energy’ which we take to be a general law of nature from
innumerable experiments. Such theories are all the more dangerous if they
accidentally explain other phenomena, as Weber’s explain induced currents" 392.
Tait again in 1868:
"But the investigations of these authors (Riemann and Lorenz) are entirely based
on Weber’s inadmissible theory of the forces exerted on each other by moving
electric particles, for which the conservation of energy is not true, while
Maxwell’s result is in perfect consistence with that great principle." 393.
Tait’s (and Thomson’s) objection refers to the contradiction between Weber’s
law and Helmholtz’s PCE, but cannot be accepted as an objection that Weber’s
law is incompatible with the impossibility of perpetual motion. The existence of a
potential for Weber’s law implies that the work done by the electric forces is a
perfect differential. This means that in a cyclical process an indefinite amount of
work cannot be generated by a particle moving under the action of the force
assumed by Weber. Thus work is not created out of nothing. Finally both
Helmholtz (in 1870 and 1872) and Maxwell (in 1873) agreed on this important
391 Helmholtz, Hermann. "Ueber die Theorie der Elektrodynamik." Berlin Monats
April 1872: 247-56. Repr. in WA 1 : 636-46. Pp.645-6.
1877. P.318.
392 Quoted in Neumann, Carl. Die Gesetze von Ampére und Weber. Leipzig: Teubner,
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
the impossibility of perpetual motion. Since this is the first root of the principle of
conservation of vis viva analysed in section 3, it is to be asserted that Weber’s
law originates directly from this root. This approach to PCE was often called the
law of potential 394.
Weber in 1871 proposed his own version of the principle of energy
conservation 395. At the end of the second part of his 1871 paper Weber answers
Helmholtz’s criticisms of 1870 about the incompatibility of Weber’s force law
with some consequences of PCE. This was the starting point of the second part of
the lengthy controversy. Weber first notes the new requirement Helmholtz
imposes on PCE. The vis viva cannot become infinite, or otherwise an infinitely
great amount of work could be performed (either in passing from a finite velocity
to an infinite, or from an infinite to a finite). Thus a limiting velocity must exist.
For Weber this limiting velocity is his constant cw=⎟2 vel.light. In Weber’s
view, Helmholtz’s criticism has to be rejected for it assumes an initial velocity of
the particles greater than c. Assuming this limiting value Weber’s potential is
always a positive quantity. In the second place, Weber notes, the finite distance
at which the particles in Helmholtz’s objection would acquire an infinite velocity
is extremely small, outside the domain of enquiry. Thus the objection is
practically meaningless. But Helmholtz again in 1872 and 1873 criticised
Weber’s law because of the negative sign in one of the terms of the generalised
potential. In fact this implies that a charge behaves "somewhat as if its mass were
negative, so that in certain circumstances its velocity might increase indefinitely
under the action of a force opposed to the motion" 396. Maxwell in 1873 agreed
with Helmholtz’s 1872 criticisms and asserted that the latter "... impossible result
is a necessary consequence of assuming any formula for the potential which
introduces negative terms into the coefficient of v2" 397. Hoppe, an historian
who defended Weber’s assumptions in two books, first in 1884 and then in 1927,
asserts that in 1875 Weber succeeded in showing that with a proper
reinterpretation of his equation for the vis viva, Helmholtz’s criticism can be
394 Neumann Amp u Web p.337.
395 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen, insbesonder uuber das
Prinzip der Erhaltung der Energie." 1871. In Werke 6 vols. Berlin, 1892-4. Vol.4. Pp. 247-99.
Tr. in Phil Mag 43 (1872): 1-20 and 119-49.
396 Whittaker Aether p.204.
397 Maxwell Treatise par.854.

efuted. But again in 1881 Helmholtz criticised Weber on the grounds of a
possible physical case in which the law would have given an imaginary value of
the velocity 398. Hoppe noted other aspects of the controversy: C. Neumann in
1871, 1875 and 1877 and Zoellner in 1876 published in favour of Weber and
against Helmholtz. Helmholtz in 1876 recognized that Rowland’s experiments on
convection were not in disagreement with Weber’s law. In 1887 Budde made a
very sophisticated analysis of the theories of Weber, Clausius and Riemann, and
planned a series of experiments "in order to prove their validity" 399. C.
Neumann’s contributions of 1877 and 1898 are extremely interesting for an
objective judgment of this long dispute. C. Neumann in fact was a defender of
Weber’s position. Still in 1877 he answered Helmholtz’s criticism of 1872
against Weber in the following terms: "The objection that follows (in Helmholtz’s
argument ) does not bear on the usual principle of energy, but on a completely
novel principle which is here pronounced for the first time. While the usual
principle of energy demands for each system the existence of an energy function,
i.e. the existence of a function which has the quality to increase in each time
interval by the same amount as is the work which is being performed on the
system from the outside - this new principle demands not only the existence of
such a function, but at the same time a very specific feature of it, in as much as it
asserts that the kinetic part of this function must invariably be positive." 400 In
Weber’s approach the electrokinetic potential has terms containing squared
velocities like the kinetic energy ones. But in contrast with standard kinetic terms
which are multiplied by a positive coefficient like m, these new terms can be
multiplied by negative coefficients, like electric charges. The sum of the resulting
positive and negative terms containing squared velocities can be negative. From
Helmholtz’s point of view, this sum represents a negative kinetic energy and thus
the new requirement of positive T was postuled . C. Neumann explicitly asserts
that physical principles are not to be considered established once and for all.
Modifications are possible and the one just introduced by Helmholtz must be
carefully analysed. C. Neumann outlines three shifts in PCE: the principle of
conservation of vis viva, the principle of conservation of energy, and Helmholtz’s
new principle of positive kinetic energy. Being a new principle, for Neumann it
cannot be considered a secure ground for debate. Helmholtz’s objections against
398 Hoppe Histoire p.591.
399 Hoppe Histoire p.592.
400 C.Neumann Amp and Web p.322.

Weber have to be analysed "apart from that principle". That is, the requirement of
positive kinetic energy as a general rule is denied, but the possibility of specific
inadmissible conclusions from Weber’s law (which admits of negative kinetic
energy) is accepted. But as to this second problem C. Neumann also remarks that
Helmholtz’s criticism can be refuted: "The fact stressed by Helmholtz, i.e. the
occurrence of infinitely great accelerations, leads to either the conclusion that
Weber’s law is inadmissible, or that the singular states are not possible. And it
would therefore be overhasty to jump to the conclusion that one alternative is
preferable over the other and to take this fact to speak against Weber’s law" 401.
In fact, in his 1877 analysis, C. Neumann admits the second alternative and
refuses the first: "Indeed, this fact would only be evidence against Weber’s law if
one could prove, in an example, that those singular states are indeed possible. We
shall see later how far away we are from succeeding to find such proof." 402. But
twenty years later, in 1898, his position was different. He finally published the
second volume on "Electric forces". The first volume had been published in 1873
and was dedicated to an analysis of AmpÏere and F. Neumann’s theories.
Following the plan of the first volume the second should have dealt with Weber’s
and Kirchhoff’s theories. But instead the 1898 volume was dedicated to
Helmholtz’s theory. Neumann had finally accepted Helmholtz's theory.
It is well known that Helmholtz’s contributions to electrodynamics were also
relevant from the experimental point of view. In fact in the light of the
equivalence of theoretical predictions of the three theories for closed currents, he
stressed the importance of experimenting with open currents. This approach was
to lead to Hertz’s experiments of the late eighties, but still in the seventies
Helmholtz suggested three important experiments to Schiller (1874), Rowland
(1876) 403 and Hertz himself (1879). The first two experiments were interpreted in
favour of the dielectric theories (whether Helmholtz’s or Maxwell’s). Rowland in
particular demonstrated the electrodynamic effects caused by convection
currents. Hertz’s experiment was to deny the relevance of Weber’s electrical
inertia 404.
401 Ibidem p.324.
402 Ibidem p.324.
403 Woodruff "Helmholtz" pp.308-10.
404 Helmholtz n.394 p.6.

But was important here is that despite the fact that, in the late seventies, no
conclusive evidence existed for a specific theory, Helmholtz came gradually to
consider dielectric theory as correct.
"In the Faraday lecture of 1881 he predicted the decline of action-at-a-distance
on the Continent and lent full support to Maxwell’s theory" 405.
But he was still convinced, against Maxwell, that electricity consisted ultimately
in discrete charges, "atoms of electricity". Thus the main element accepted from
Maxwell was the role of the dielectric and, gradually, contiguous action.
Helmholtz’s shift towards contiguous action and, in particular, the relation of the
shift with the new version of PCE (Poynting's principle of local conservation)
deserves detailed attention. Helmholtz’s views at the beginning of the eighties
had already undergone some changes. His 1847 Newtonian model of central
forces depending only on positions was changed in 1870 into a model of forces
and torques. The theory of the dependence of forces on the positions was also
lost, when Helmholtz accepted the general electrodynamic potential, a concept
first introduced by F.Neumann, Weber and Clausius. In 1872 Helmholtz asserted
the further condition on PCE, i.e. that the generalised kinetic energy of the
electrical system (deriving partly from electrodynamic potential) must always be
positive. Very few aspects of Helmholtz’s early conception remained at the
beginning of the eighties. One of these was the distinction between kinetic and
potential energy. While shifting towards contiguous action, a conception that in
the end would have denied this distinction, Helmholtz underwent a second,
almost contemporary, shift: the Principle of Least Action was now to be
considered the key to physical research, more than the Principle of Conservation
of Energy.
Unification, again ( 1884-94)
In a long paper of 1886, Helmholtz introduces a new unifying
programme in physics, based on a heuristic principle different from energy
conservation : the principle of least action (PLA). The principle of conservation
of energy is abandoned as the main guide in physics :
"it in fact holds in a great variety of cases when least action does not.
The latter is thus more specific".
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
expression of the kinetic potential in Maxwell's theory. The old restrictive
hypothesis according to which velocity is solely part of the vis viva, as a
homogeneous quadratic function, has to be dropped. Helmholtz, at the very
moment when the contiguous action theory was receiving its own non
mechanistic local principle of conservation, gives up with the requirement so
strongly defended for almost forty years: he recognizes that energy can be any
function whatever of the coordinates and velocities. The retreat from the 1847
positions is now almost complete: a lasting element is that the new interpretation
of energy is still in the mechanical framework.
But Helmholtz, now sixty-four and at the height of an extraordinary
career, is again ready to start working towards new unifying horizons. The new
regulative principle, PLA, should not be confined to mechanics and optics;
Helmholtz applied it to electrodynamics and attempted to extend its range to
thermodynamics:
"From these facts we may even now draw the conclusion that the
domain of validity of the principle of Least Action has reached far beyond the
boundaries of the mechanics of ponderable bodies. Maupertuis’ high hopes for
the absolute general validity of his principle appear to be approaching their
fulfilment, however slender the mechanical proofs and however contradictory the
metaphysical speculations which the author himself could at the time adduce in
support of his new principle. Even at this stage, it can be considered as highly
probable that it is the universal law pertaining to all processes in nature ..... In
any case, the general validity of the Principle of Least Action seems to me
assured, since it may claim a higher place as a heuristic and guiding principle in
our endeavour to formulate the laws governing new classes of phenomena." 406
These statements are part of the 1886 paper on PLA. A second paper
was to appear in 1887. C. Neumann analysing Helmholtz’s work in a long
treatise of 1898, remarks that Helmholtz's electrical investigations can be divided
into two groups: the first (1870-75) has its roots in the general conceptions of
Newton’s gravitational theory, i.e. in the principle of direct actionat-a-distance.
406 Helmholtz, Hermann. "Ueber die physikalische Bedeutung des Princips der
kleinsten Wirkung." In Crelle's Jour 100 (1886): 137-66 and 213-22. Repr. in WA 3. Pp.203-
. Quot. from; Yourgrau and Mandelstam. Variational Principles in Dynamics and Quantum
Theory. 2nd ed. London: Pitman, 1960. P.143.

The second (1892-94), follows the investigations of Faraday, Maxwell and Hertz
and shares its general basic features with Fourier’s theory of heat conduction.
"Like the latter it rests on the idea that the true origins of any changes
occurring in any point of the universe must be found in the immediate vicinity of
this point" 407.
It is thus interesting to analyse the reasons for this double almost
contemporary shift: from action-at-adistance to contiguous, from conservation of
energy to least action, and their possible connection. The analysis is here outlined
only with reference to the modification of the concept of energy and to the
expression of PLA in the ’86 paper.
Helmholtz recognizes the following points:
a) H=F-L and E=F+L (where H is the Hamilton function, F potential
energy and L vis viva); the name of kinetic potential should be attributed to H,
after F. Neumann’s work on mutual potential, and Clausius’ work on
electrodynamic potential. As already recalled the old restrictive hypothesis
according to which the velocity is solely part of the vis viva, as a homogeneous
quadratic function, has to be dropped. H can be any function whatsoever of
coordinates and velocities.
b) In the electrodynamic case such functions have been given by F.
Neumann, Clausius and Weber. But the dielectric has to be taken into account,
following recent results. In this case, the form of the function differs from that for
ponderable masses.
c) The starting point for the research of the ’86 paper was the search
for an expression of the kinetic potential in Maxwell’s theory.
d) In a great variety of cases, the conservation of energy holds, while
least action does not. Thus PLA expresses a particular feature of the conservative
forces not intrinsically connected with their conservative character. That is, it
says something else which is more specific 408. In view of these remarks, my
opinion is that Helmholtz’s shift towards Maxwell in the late seventies was due
to the possibility of localising the energy in the dielectric. This allowed for a
better explanation of optics, of experimental results and also for a distinction
between kinetic and potential energy of the dielectric medium to be maintained,
while in the other theories this distinction was lost. At the same time, most of the
407 Neumann, Carl. Die Elektrischen Kräfte. Part 2. Leipzig: Teubner, 1898. P.IV.
408 Helmholtz "kleinst Wirk" Introduction and par.2.

different competing theories showed an agreement with some versions of PCE,
mainly with the impossibility of perpetuum mobile. That is, they admitted a
potential. Thus, strictly speaking, there was no longer any heuristic power in the
PCE (still a global one in Helmholtz’s view). Either a more specific version of
PCE or a more specific regulative principle was needed. The more specific
version of PCE was to be connected with the localisation of energy of Poynting,
but Helmholtz seems not to have understood its relevance, the more specific
principle was in Helmholtz’s view PLA. Moreover the PLA version had to be in
agreement with contiguous action: in fact a PLA related with Maxwell’s theory
was Helmholtz’s starting point in 1886. At this stage the two shifts had been
accomplished before Hertz’s experiments.
In 1894 Helmholtz publishes a small volume entitled "Introduction to
the Lectures on Theoretical Physics". The first two sections, Introduction and
Part I, are closely related to the themes discussed fourty years before in the
Erhaltung and with them we will be concerned here. The third section, Part II, is
very similar to the paper "An Epistemological Analysis of Counting and
Measurement".
Introduction and Part I are of great relevance to focus some aspects of
Helmholtz's methodological beliefs. Namely the conditions of possibility of
understanding nature, the relations betwen force and law, the role of regulative
principles in framing our scientific knowledge.
What had been achieved: a turn of the century viewpoint
Energy theory changed the shape of physics : after 1847 physical laws
no longer had only to face the challenge of experimental results, but also had to
be judged on more theoretical grounds, in their relation with the conservation
principle. But in turn the principle had, from the beginning, different
formulations.
The growth of these formulations was joined with the growth of
alternative research programs in the second half of the century. The energetist
movement took off, with its attempts to overthrow the mechanical worldview,

while the electromagnetic and the statistical approach, both deeply connected
with energy problems, were actively pursued. Since energy conservation became
one of the basic chapters of physics, the champions of the different research
programs dedicated a great deal of logical and historical analysis to an
understanding and framing of the various contributions and developments.
Helmholtz was a leader in this spread of theoretical physics in the second half of
the century, and is interesting to find out what was the judgement of his fellow
scientists on his approach to PCE.
I have already dealt with the first debate on the history and role of the
energy pioneers, but besides Tyndall (1863) and Tait (1868 and 1876), from the
seventies a number of books appeared on the history and foundations of energy
theory which, while demonstrating the now recognized importance of the subject,
gave interesting assessments of Helmholtz's contributions. Among them 409:
Maxwell (1870 and 1877), B.Steward (1874), Stallo (1882), Lodge (1929), and
in Germany : Mach (1872, 1883 and 1896), Planck(1887), Helm (1887 and
1898), Ostwald (1903 (tr fr.1912) and 1908), Haas (1909), and in France :
Duhem (1895,1905), Poincarè (1892 and 1902). Rowland (1882) analysis of the
experimental determinations of the mechanical equivalent of heat is also
interesting.
Philosophers too at the beginning of the century took a great interest in
evaluating the role of Helmholtz's formulation of energy conservation, among
these : Meyerson (1907) and Cassirer (1910). Important among the historians is
the contribution of Mertz (1965 rep), that anticipates some modern
historiographical claims.
I will confine myself here at discussing some remarks on Helmholtz's energy
theory and on his specific methods made by Planck, Helm and Poincarè.
Planck outlines a basic problem in the formulation and application of the
principle of conservation of energy, in whichever form : it is impossible to find
the primary expression of the energy 410; the substantialization of energy is open to
a certain degree of arbitrariness 411; there is a difficult theory/ experiment
interplay 412. These general remarks are valid for all the expressions of the
principle. What are the merits and demerits of Helmholtz's specific formulation?
409 See n.7.
410 Planck Prinzip p.114.
411 Planck Prinzip p.104.
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
great merit, a simplification and a great heuristic tool, but that nevertheless this is
insufficient to deal with in complicated situations, like electromagnetism, still
partly unknown. This means that the principle 'per se', without the contribution of
experimental results, cannot give definite answers.
To apply Helmholtz's's principle to a process that takes place in a system of
bodies it is necessary to add together all the different kinds of variations of vis
viva on one side, the sum of tension forces on the other side and equate the two
groups. The sum of all the identified terms representing the vis viva and the
tension forces at a given instant is the "force" (energy) of the system, constant if
the system is isolated. Of course all the identified different terms of the sum have
to be numerically expressed on the basis of a common unity of measurement,
usually mechanical work; thus their mechanical equivalent has to be found. But in
this process of identification of the different terms contributing to the energy of
the system we find a methodological difficulty:
"there is not a general rule with which we can calculate a priori the value of the
equivalent (i.e. the expression of the different terms), independently of the
principle itself" 413
Often in the past wrong identifications of the equivalents led to wrong
conclusions (Planck quotes Descartes and Carnot 414).
Helmholtz's principle with its reduction of all the energy terms to two main forms,
kinetic and positional, is without doubt a great heuristic tool to find out the
different energy terms, but does not solve all the problems. That is, there is
always an indetermination and then an arbitrariness in the identification and
expression of the energy terms.
There is in fact a complicated interplay between the general framework of the
principle and the empirical laws.
From an existing and accepted empirical law the expressions of the tension
forces and of the vis viva have to be identified (and the law rededuced). Instead
for a realm of phenomena not yet described by laws or with laws not sufficiently
corroborated, if we assume the principle to hold true some useful guide-lines for
the formulations of empirical laws can be produced. Sometimes the principle
leads to discover new laws, in other cases the discovery of new experimental
laws will add new energy terms to the principle. Again new experimental data
413 Planck Prinzip p.38
414 Planck Prinzip Pp.7-9 e 13-17

can modify its specific theoric formulation, but not its validity, that Planck
considers true as far as numerical results are concerned 415. Following Planck in
fact the numerical value of the work equivalent of a transformation of a system
between two states is always the same, independently of the way in which the
transformation takes place. This he considers as an experimental result. But the
theoretical interpretation of the energy terms can be achieved in different ways in
different theories: the process of "substantialization" is open to a certain
arbitrariness, always useful for the progress of knowledge 416. Planck ends his
1887 analysis pointing to one of this progress, the overcome of the action at a
distance approach in electrodynamics, based on the (theoretical) success of the
new version of the principle of conservation: the local conservation of energy.
There is not a unique way to apply the general framework and thus a great part of
the success relies in the ability of the scientist to use this new powerful
theoretical tool: thus Helmholtz's 1847 success is due not only to his theoretical
formulation but to his widespread and deep knowledge of the most various
branches of natural science, a fact often overlooked by commentators. Only when
his empirical knowledge fails, as in the case of electromagnetism, the
methodological problems of the theory/experiment interplay in the formulation
and application of the principle become evident.
So far Planck's warnings. A different approach is outlined by Helm: of the two
roots of Helmholtz's principle he accepts the first one (impossibility of perpetual
motion) and denies the validity of the second (central forces). Thus Helm, while
accepting a principle of conservation and a concept of energy, refuses as
unnecessary Helmholtz's two main energy forms: tension and living forces. For
Helm there is no theoretical nor practical reason to specify in an energy balance
which is a tension force and which a living one, given that what matters is the
work equivalent of a specific energy term. Helm denies 417 both the central forces
hypothesis and the mechanical view of nature. Energy conservation means energy
correlation and the equations are nothing else that sums of work equivalents. The
mechanical view of nature, Helm asserts in line with Mayer's approach, does not
add anything to the constancy of the sum of the work equivalents. I want to
underline that the refusal of the privileged status of the central forces does not
necessarily imply the rejection of the mechanical view: Clausius, as discussed
415 Planck Prinzip p.99, see also the Introduction to the 1887 edition.
416 Planck Prinzip Pp.104-5.
417 Helm Energetik p. 41.

above, accepted the mechanical view and rejected the limitation to central
forces and thus the distinction between positional and kinetic energy terms. For
Clausius the specific condition needed to fulfill energy conservation is that the
work done by the forces has to be a total differential 418.
Much different the position of another great scientist: Poincarè. Helmholtz clear
distinction between potential and kinetic energy is for Poincarè basic to give a
real meaning to the general concept of energy. From a mathematical point of view
there are many invariants in a transformation and without Helmholtz's criteria
there would be serious difficulties in making a choice. Thus Poincarè 419 accepts
completely Helmholtz's approach and believes that this is the only solution to
solve the difficulties connected with the arbitrariness of the energy concept.
In 1902 Poincarè published a synthesis of his works in La Science et
l'Hypothèse , a famous book, important here because it shows that energy
considerations were still fundamental for the scientific debate. The first edition of
the first volume of Poincarè’s Electricitè et Optique was published in 1890. It
collects the lectures given between March and June 1888 (and not 1889 as
erroneously printed on the volume). The whole book concerns Maxwell’s theory.
Poincarè’s Introduction became very famous, for he analyses the difficulties a
French reader has to face reading Maxwell’s Treatise . The difficulties are
related to a lack of precision and logical order, a lack which conceals Maxwell’s
main result. This main result plays a great part in Poincarè’s analysis of the CET
debate. Which is Maxwell’s main result?
"Pour dèmontrer la possibilitè d’une explication mècanique de l’èlectricitè, nous
n’avons pas à nous prèoccuper de trouver cette explication elle-meme, il nous
suffit de connaitre l’expression de deux fonctions T et U qui sont les deux parties
de l’ènergie, de former avec ces deux fonctions les equations de Lagrange et de
comparer ensuite ces èquations avec les lois expèrimentales." 420
Several points are contained in this statement: the necessity for a mechanical
explanation of electricity, the sufficiency of the possibility of the explanation, the
role of a PCE with a sharp distinction between T and U, the link of a Lagrangian
derivation with this PCE. The actual kind of mechanical explanation is not
1968.
418 See above.
419 Poincaré, Henry. La Science et l'Hypothèse . Paris, 1902. Rep Paris: Flammarion,
420 Poincarè, Henry. Electricité et Optique. I Les Theories de Maxwell. Paris: Carré,
1890; pp.XIV-XV.

considered relevant. In Poincarè’s view, the demonstration of the possibility of a
mechanical explanation i.e. the fulfillment of Lagrange equations, is the main
point in Maxwell’s analysis. This allows him to accept the difficulties and
contradictions that sometimes blemish the British masterpiece. In my analysis,
Poincarè’s remarks are fundamental for two reasons: first, because they show the
importance of PCE and PLA, i.e. of regulative principles, as grounds of debate;
second, because they show that Maxwell’s merit was considered to be the
application of Helmholtz's PCE, a specific PCE with a sharp distinction of T and
U. Thus Poincarè is not concerned, unlike Planck, with the local conservation in
Poynting’s sense, but still judges Maxwell's theory on the basis of Helmholtz's
PCE, the same that Maxwell adopts. For Poincarè in 1890, the sharp distinction
between T and U is still very important, and we are going to see that he will
consider it basic in later works too. But still this is not Poincarè only meaning of
conservation: in 1890 he asserts that a consequence of the assumed conservation
of energy is the existence of a force function (= - U, where U is the potential
energy) so that the equations of movement can be expressed as 421:
etc. Sometimes Poincarè asserts that one of the expressions of conservation of
energy is the existence of a potential depending only on positions 422; in the case
of two electric circuits the terms expressing an exchange of energy are:
where is the electric energy provided by two batteries;
is the Joulean heat produced in the circuit and dT is the part of the variation of
the electrodynamic potential in itself of the system of the two circuits due to the
displacement of the circuits. Now PCE asserts that the previous expression has to
be zero for a closed cycle or to be an exact differential in other cases 423. In this
last example, reference to potential and kinetic energy in the expression of PCE is
avoided. Other relevant passages of this first edition are the statements of the
existence of two electrostatic theories deducible from Maxwell’s and that in the
first theory electrostatic energy cannot be considered as potential 424. Instead
421 Poincarè El et Opt p.X.
422 Poincarè El et Opt p.117 (par.106).
423 Poincarè El et Opt p. 165 (par.149).
424 Poincarè El et Opt p.92 (par.84).

Poincarè agrees that the electrodynamic potential of a system of currents is the
kinetic energy of the ether 425.
In contrast to the great interest in energy theories, little room is left for Hertz’s
experiments.
Analysing Weber’s theory, Poincarè asserts that it agrees with PCE, as far as the
work of electrodynamic repulsion is equal to the differential - dc of the potential
c. 426 In this case Poincarè is using the potential law as PCE, but somewhere else
he adopts Helmholtz’s PCE, and clearly refers to the distinction between kinetic
and potential energy. In the analysis of Helmholtz’s theory the expression of PCE
is the following: the variation of T+U must be equal to the work performed by the
external electromotive forces (chemical, thermoelectrical, etc.) minus the Joulean
heat 427. The need for the distinction of T and U appears relevant in the subsequent
analysis of the different theories: Weber’s is rejected, despite its fulfilment of
PCE, because of the negative value of the constant K in Helmholtz’s general
potential. In fact K < O leads to negative kinetic energy and instabilities 428. The
relevance for Poincarè of PCE and, moreover, his specific preference for a PCE
involving T and U has already been outlined. His preference for Maxwell was a
result of this choice. Equally remarkable is that Poincarè’s made a link between
PCE (with T and U sharply divided) and the Lagrangian derivation: he never
considered the possibility of a Lagrangian derivation with an electrokinetic
potential, and thus refers the possibility of a mechanical explanation (given by the
Lagrangian derivation) to the possibility of dividing T and U.
Poincarè’s ideas on energy are clarified in a work of 1892 429. The principle of
conservation is called here Mayer’s principle and Poincarè wonders at the
success it has among other physical laws. A reason cannot be found in its
connections with the impossibility of perpetual motion (from which it can be
derived only in the case of reversible phenomena). Moreover the attempts at a
clear definition are useless: "it is impossible to find a general definition of it". The
principle disappears when generalised and applied to the universe. The only
425 Poincarè El et Opt p.169 (par.152).
426 Poincarè, Henry. Electricité et Optique. II. Les Theories de Helmholtz. Paris:
Carré, 1891; p.31, 41 (par.18-20).
427 Poincarè El et Opt 2 p.69 (par.31).
428 Poincarè El et Opt 2 p.75 (par.34).
429 Poincarè, Henry. Thermodynamique. Paris: Carré, 1892; p.IX. Repr. in Poincarè
La Science pp.144-9.

meaning left is: "There is something which is constant". But what is this
something? Poincarè distinguishes two cases: a universe whose evolution is
completely determined by the values of n parameters and of their derivatives and
a system in which there are p of the n parameters that vary independently (that is,
the system is a limited one, interacting with the exterior world). In the first case,
the n existing differential equations admit n-1 first integrals, i.e. constants. It
would be difficult to decide which deserves the name energy. Thus the principle
is nonsense. In the second case, the principle expresses a limitation: the n-p
relations admit a combination whose first member is a complete differential; if the
work of the external forces and the heat exchanged is zero, the integral of the
resulting zero differential is a constant, i.e. the energy. 430 Here conservation of
energy as existence of a complete differential, is clearly defined, but with the
constraint of admitting a limited system interacting with the outside. At this stage,
Poincarè does not discuss the Helmholtzian interpretation of energy as the sum of
T and U, but it will be seen that this was to be his choice, in order to avoid the
difficulties just expressed.
In 1902 Poincarè improves his analysis: now again, as in 1890, the distinction
between T and U is fundamental to give a specific meaning to energy. In an
analysis that became famous, Poincarè showed that, to give a specific meaning to
energy, its terms have to be of a particular form, one depending on square
velocities and one on positions 431. Weber’s case is explicitly quoted: a distinction
between kinetic and potential energy being impossible, how can we define
energy? 432 In this analysis, where PCE is now called Helmholtz’s and not
Mayer’s principle, the cases in which energy cannot clearly be divided into T and
U, (when for instance also an internal energy Q appears, not clearly distinguished
from the other two), are considered tautological expressions of conservation:
something is conserved, but the whole assertion is untestable due to its
unspecificity. Poincarè is thus led to what Planck called a substantialisation, and
the substantialisation is clearly expressed in the electromagnetic case. First of all
the distinction between T and U. Second their interpretation in electromagnetic
terms and their localisation:
"Et alors Maxwell s’est demandè s’il pouvait faire ce choix et celui des deux
ènergies T et U, de facon que les phènomenes èlectriques satisfassent à ce
430 Poincarè Thermod p.XI. Repr. in Poincarè La Science p.147.
431 Poincarè La Science pp.139-44.
432 Poincarè La Science p.141.

principe. L’expèrience nous montre que l’ènergie d’un champ èlectromagnètique
se dècompose en deux parties, l’ènergie èlectrostatique et l’ènergie
èlectrodynamique. Maxwell a reconnu (que si l’on regarde) la premiére come
reprèsentant l’ènergie potentielle U, la seconde comme reprèsentant l’ènergie
cinètique T." 433
Maxwell thus solves the problem, allowing in addition a Lagrangian derivation of
the equations and so fulfills the possibility of a mechanical explanation of
electromagnetism. The grounds of acceptance of Maxwell’s theory in Poincarè’s
analysis are clearly the ones referring to energy. Maxwell’s (and not Poynting’s)
approach is preferred in view of its links with the mechanical expression of both
PCE and PLA. This judgment is explicit already in 1890 and was to be reasserted
in 1902 434. Hertz’s experiments, instead, up to 1894 were considered doubtful
and in 1902 were considered to lend only indirect support. The
desubstantialisation of ether theory was to weaken the experimental difference
between contiguous action and delayed action-at-a-distance theories, but was to
strengthen the theoretical difference between the PCE’s of the latter theories and
a local PCE expressed by the contiguous action theory. Poincarè’s choice of
Maxwell from 1890 to 1902 is not referred to Poynting’s development (second
step of localisation) but to Maxwell’s substantialisation (first step). This is due to
the relevance he attributed to the asserted possibility of a mechanical
explanation 435. Poincarè’s approach is very different from Hertz’s axiomatic
assumption of Maxwell’s equations and his rejection of both a mechanical
explanation and of a Lagrangian derivation. But still PCE, and specifically
Helmholtz's PCE, was the principal reason for Poincarè’s choice of Maxwell.
In the important process of the emergence of theoretical physics
discussions on the meaning and the role of Helmholtz's approach to energy
conservation were basic. Acceptance or denial of the mechanical world view, of
the sharp distinction between kinetic and potential energy and of the privileged
status of the central forces depending only on the distance were among the key
issues, while others, non strictly mechanical interpretations of the principle of
conservation, started to become mere and more important. At the end of the
century in every research programme (mechanical, electromagnetic, energetic,
thermodynamic) a specific version of the principle was put forward, and new
433 Poincarè La Science p. 223.
434 Poincarè La Science pp.216-25.
435 Poincarè La Science pp.216-25.

interpretations at the beginning of the new century were to be proposed in the
new programmes of relativity and quantum physics. The theory-experiment
interplay did not enjoy any longer a privileged position: the theory-principle one
was now at the forefront of research, showing the lasting importance of
Helmholtz's 1847 methodology. It is significant that the emergence of theoretical
physics was joined with a deep work on the history of physics, with approaches
that often were methodologically sophisticated, even for today's standard.
Contemporary Historiography
New trends in energy studies cannot avoid discussing what without doubt
started the recalled fourth phase in the debate on the history of energy
conservation: Thomas Kuhn's paper on the "simultaneous discovery" of "energy
conservation" 436, published in 1959. Kuhn's paper is still unanimously defined as
challenging, for the lack in contemporary historiography of a rival synthesis 437.
Nevertheless analysing Helmholtz's Erhaltung , and the related primary and
436 Kuhn, Thomas. "Energy Conservation as an Example of Simultaneous Discovery."
In Critical Problems in the History of Science . M.Clagett ed. Madison: Wisconsin U.P.,
1959. Pp. 321-356. Rep. in Kuhn, Thomas. The Essential Tension, Chicago: Chicago U.P,
1977. Pp. 66-104.
437 Cantor, Geoffrey. "Locating the First Law of Thermodynamics." In New
Perspectives in Nineteenth-Century Science. University of Kent, 1984.

secundary literature, I started doubting of the correctness of some of Kuhn's main
historical and historiographical claims.
Six not minor points are, in my view, at issue: 1) that in 1845-46
"Helmholtz fails to notice that body heat may be expended in mechanical
work" 438; 2) that the concept of work of the old mechanical engineering tradition
was "all that which is required" and "the most decisive contribution to energy
conservation made by the nineteenth-century concern with engines." 439; 3) that
"Helmholtz was not, however, aware of the French theoretical engineering
tradition. Like Mayer he derives the factor of1/2 in the definition of energy of
motion and is unaware of any precedent for it." 440; 4) that Helmholtz "fails
completely to identify as work or Arbeitskraft and instead calls it the "sum
of the tensions.." 441; 5) that "the dominance of contact theory in Germany" might
"account for the rather surprising way in which both Mayer and Helmholtz
neglect the battery in their accounts of energy transformations" 442; 6) that
"Helmholtz was able by 1881 to recognise important Kantian residues " in the
Erhaltung "that had escaped his earlier censorship" 443 and that this is evidence of
influences of Naturphilosophie .
I will show below that: 1)Helmholtz before 1847 was aware of the heatwork
interconvertibility, despite lacking a numerical equivalent; 2) the concept of
work was not a precious contribution to energy conservation till it acquired the
requirements of being a total differentiaL; 3) Helmholtz was well aware of the
French tradition; he utilised but did not rederive the new expression of the vis
viva; 4) Helmholtz consciously dropped also the new interpretation of the term
"Arbeit" in favour of "Spannkraft". This was one of his most relevant
contributions. 5) Helmholtz not only showed that contact theory was not against
conservation, but also dedicated the longest section of the Erhaltung to an
analysis of the batteries; 6) Helmholtz did not censor but reinstated in 1847 the
philosophical introduction to the Erhaltung before publication were Kantian
transcendentalism and not Naturphilosophie idealism played a main role. In 1881,
438 Kuhn Sim Disc p.95 n. 68.
439 Kuhn Sim Disc p.84 and 90.
440 Kuhn Sim Disc p.88.
441 Kuhn Sim Disc p.88.
442 Kuhn Sim Disc p.73
443 Kuhn Sim Disc pp.100-1.

while fighting metaphisicians, he was still stressing Kantian aspects of his earlier
approach.
What is more important some major aspects of Helmholtz's Erhaltung
escaped Kuhn's (and other historians') attention: 1) Helmholtz's methodological
four level structure; 2) his demarcation between theoretical and experimental
physics; 3) his overcoming of both the engineering and the mathematical
approach to the work concept; 4) the lack of an experimental determination of a
work equivalent of heat and the mistranslation of Joule's values; 5) the difficult
(and sometimes wrong) theory-experiment interplay in the application of the
principle; 6) the formulation of a lasting methodology and of a non lasting
conceptual model of energy.
All that in my view leads to a historiographical problem: in fact if history
does not get clarified "perhaps.. the wrong.. questions have been asked" 444
A closer look at Kuhn's paper is needed: Kuhn claimed that twelve
scientists, divided into three groups of four 445 , between 1832 and 1854
"grasped for themselves" essential "elements" of the concept of energy and of its
conservation. The paper addresses the problem of why these "elements" became
accessible in those two decades and seeks to identify not all the "prerequisites"
which resulted in the theory of energy conservation but only the "trigger factors"
specific to the period. Kuhn outlines three: the "availability of the conversion
processes", the "concern with engines" and the "philosophy of nature" 446.
In my view two major historiographical problems are related to Kuhn's
use of the terms "simultaneous discovery" and "energy conservation" 447. The first
444 Cantor, Geoffrey. "William Robert Grove, the Correlation of Forces and the
Conservation of Energy." In Centaurus 19 (1976): 273-90. P.287; see also P.273.
445 a) combining generality of formulation with concrete quantitative applications,
between 1842 and 1847: Mayer, Joule, Colding and Helmholtz; b) asserting the
quantitative interchangeability of heat and work and computing a value for the
conversion coefficient, between "before" 1832 and 1854: Carnot, Séguin (1839),
Holtzmann (1845) and Hirn (1854); c) believing in a single force that in all its
transformations can never be created or destroyed, between 1837 and 1844:
Mohr, Grove, Faraday, Liebig. Kuhn Sim Disc pp. 66-9.
446 Ibid p.73
447"energy conservation" is supposed to be the most "striking instance" of a
simultaneous discovery. Kuhn Sim Disc p.69.

term derives from a sociological debate 448 challenging the "priority" approach 449.
Kuhn, adopting this sociological perspective, is obliged to identify different
statements of the pioneers 450 with a "simultaneous discovery", despite some
explicit difficulties with a stricter historical perspective 451. Paradoxically he gives,
against his own basic historiographical claims 452, a Whiggish explanation 453.
Whatsmore in so doing, and here we deal with the second historiographical
problem, he adopts a concept of energy that has substantialistic tones 454 and that
is by no means shared contemporary physical knowledge 455. Nowhere in the
paper can be found an explicit definition of "energy conservation", a rather
serious weakness, shared by the great majority of the contemporary secondary
448 "the main objective" of the paper is the preliminary identification of the sources of
this phenomenon. Kuhn Sim Disc p.70. Elkana, Yehuda. "The Conservation of Energy: A case
of Simultaneous Discovery?" In Arch Int Hist Sci 24 (1970): 31-60. Pp 36-7.
449 Kuhn Sim Disc n.8, p.72. See also: Cantor "Locating" p. 1.
450 "no two of our men even said the same thing". Kuhn Sim Disc p.70.
451 "Even to the historian acquainted with the concepts of energy conservation, the
pioneers do not all communicate the same thing.", and " What we see in their works is not
really the simultaneous discovery of energy conservation". Kuhn Sim Disc p.72.
452 Against the scientist-historians who "retrieved the current contents of the specialty
from past texts of a variety of heterogeneous fields, not noticing that the tradition they
constructed in the process had never existed. In addition, they usually treated concepts and
theories of the past as imperfect approximations to those in current use, thus disguising both
the structure and integrity of past scientific traditions." Kuhn, Thomas. "The Relations
between History and the History of Science." In Daedalus 100 (1971) : 271-304. Rep. in Kuhn
Ess Tension. Pp. 127-161. P.149. Kuhn himself warns against the "inappropriateness of our
concept of discovery", as remembered by Cantor Locating p.2. See Kuhn, Thomas. "The
Historical Structure of Scientific Discovery." In Science 136 (1962) :760-64. Rep. in Kuhn Ess
Tension. Pp 165-77.
453 "Only in view of what happened later can we say that all these partial statements
even deal with the same aspect of nature". Kuhn Sim Disc p.70.
454 "We know why these elements were there: Energy is conserved; nature behaves
that way". Ibid. p.72.
455 "It is important to realize that in Physics today, we have no knowledge of what
energy is.". Feynman, Richard. The Feynman's Lectures on Physics. 3 vols. Addison Wesley,
1963. Vol.1, chapt.4, par 4.1, p.4.2.

literature 456. There is an intrinsic difficulty in assuming an implicit definition of
energy conservation as a historiographical tool: there were many formulations at
the mid of last century, many competing views in the debates of the following
decades, and even competing views in this century on the general validity of the
principle 457. Thus what is needed for an historical analysis is an explicit point of
view that does not offer "the formulation" but a possibility of grouping and
comparing the various expressions. An important contribution was already given
in 1887 in one of the "classics" mentioned above; Planck in fact clarified the
difficult meaning of the principle of energy conservation: following W.Thomson
he defined "energy" as the amount of work that can be done between two states
of a system, he asserted that can be experimentally proved that this (quantity of)
energy is conserved, he outlined a certain arbitrariety in the theoretical expression
of this quantity, he warned 458 about the impossibility of a primary definition of
energy and about the everlasting theoretical-empirical interplay that leads to
different formulations and applications of the principle. If we accept that "energy"
is not a thing, nor a theory but that it is a relational term 459 whose principle of
conservation, if adopted, had and has a number of different formulations we
understand the reason for the difficulty in the historians' works to offer "one"
explicit definition: there are in fact many. Contemporary textbooks, adopting a
superposition principle, list one for every field of inquiry (mechanical,
electromagnetic, nuclear,...) but for each of these at the time of the original
456 When a definition is referred to, there is always a lack of generality; see for
instance: Heimann, Peter. "Conversion of Forces and the Conservation of Energy." In
Centaurus 18 (1974):147-61, p.148 refers to Rankine 1853. Hiebert holds a factorisation
approach: Hiebert, Erwin. "Commentary on the Papers of Thomas Kuhn and I.Bernard
Cohen." In Critical Problems in the History of Science . M.Clagett ed. Madison: Wisconsin
U.P, 1959: 391-400. P.392; see also: Hiebert, E. Historical Roots of the Principle of
Conservation of Energy University of Wisconsin:1962. Rep: New York: Arno Press, 1981.
Pp. 1-6 and the related criticisms of Jammer, Max . "Factorisation of Energy". In BJPS 14
(1963-4): 160-6.
457 Pauli, Wolfgang. Aufsätze und Vorträge über Physik und Erknenntnistheorie
(1933-58) . Chapt.16.
the "classics".
458 Planck Princip Pp. 104-115.
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
what happened at the middle of last century the historiographical categories of
"simultaneous discovery" 461 of "energy conservation".
As a consequence Kuhn's three "trigger" factors have to be discussed as
well. Dealing with the first factor, "the concern with engines", Kuhn focusses on
the concept of work and on three traditions related to it: the older engineering
practice 462, the analytical tradition 463, dealing with vis viva conservation, where
the stress was on what was later to be called the potential function, a theoretical
engineering tradition starting with Lazare Carnot 464. Kuhn discards the last two 465
and asserts that the older tradition is the one that was really effective 466. This
460 For different versions of the principle in the history of electromagnetism see
Bevilacqua, Fabio. The Principle of Conservation of Energy and the History of Classical
Electromagnetic Theory. Pavia: La Goliardica Pavese, 1983.
461Winters criticises the same passage of Kuhn but with a different example: the
evolution of Helmholtz's ideas on energy shows that there was not a " conservation of energy"
to be discovered. Winters, Stephen. "Hermann von Helmholtz's Discovery of Force
Conservation." Dissertation. The John Hopkins University, 1985. Pp. 11-12.
462 "a century of engineering practice where its use had been quite independent of
both vis viva and its conservation". Kuhn Sim Disc P. 84.
463 Here the concept of work was not recognized as a conceptual entity: "the integral
of a force times differential path elements occurs only in the derivation of the {vis viva}
conservation law. The law itself equates vis viva with a function of position coordinates". Ibid
P.86.
464 Here work is the "fundamental conceptual parameter". Ibid. P.87.
465 For the analytical tradition see ibid. P.83 (on this point he was immediately
criticised by Hiebert: Hiebert "Commentary" P. 393). For the theoretical engineering see ibid.
P.88: "then almost none of the pioneers used it. Instead they returned to the same older
engineering tradition in which Lazare Carnot and his French successors had found the concepts
needed for their new versions of the dynamical conservation theorem.". Also : "That source
within the engineering tradition is all that the pioneers of energy conservation required and as
much as most of them used". Ibid. P. 84.
466 Ibid. p.90. Hiebert Commentary P.394 underlines that a first concept of work
meant to explain the five simple machines goes back to Hero's of Alexandria Mechanica. But
according to Cassirer the connection of the concept of work with "energy" conservation
history goes back to Leibniz. Cassirer, Ernst. Leibniz' System in seinen wissenschaftlichen
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,
approach had problems with the conservation of work 467. Only a fourth tradition,
not mentioned in Kuhn's paper, of scientists dealing with the physicomathematical
potential theory, identified the concept of work with the one of
potential 468, opening the way for a mathematical expression of "energy"
conservation. Thus if the concept of work is taken, as Kuhn does, in a loose
sense it could derive from anybody from Hero of Alexandria to Leibniz more than
from the engineering tradition of the 18th century; in any case it would be much
more a prerequisite than a trigger factor. A more stringent trigger factor or, better,
a close influence is linked with a more technical concept of work, is connected
with the emergence of the potential theory and is strictly related to vis viva
conservation. The French theoretical engineering tradition mentioned by Kuhn
has always received special attention: Ruhlmann 469 dedicated two volumes of
history to the subject, Hoppe already noted 470 the influence of Carnot on
Lagrange, Auerbach 471 is aware of this tradition, Cassirer 472 even discussed
Poncelet's approach to projective geometry in a philosophical context. Helm
Erkenntnisproblem in der Philosophie und Wissenschaft der Neueren Zeit , Berlin: B.Cassirer,
vol 2.
467 Helm Energetik P.12. Haas Entwickl. P.81. See also: Grattan-Guinness, Ivor.
"Work for the Workers: Advances in Engineering Mechanics and Instruction in France, 1800-
1830". In Annals of Science 41 (1984):1-33. P. 32.
468 Wise noted the lack of mathematical factors in Kuhn's paper: Wise, Norton.
"W.Thomson's Mathematical Route to Energy Conservation: a Case Study of the Role of
Mathematics in Concept Formation." In HSPS 10 (1979) : 49-83. P. 5O, and attributed to
Poisson the joining of the concepts of work and potential: P.64; but a different view is
espressed in Wise, Norton and Smith, Crosbie. "Measurement, Work and Industry in Lord
Kelvin's Britain." In HSPS 17 (1986): 147-73, P.154.
469 Rühlmann Maschinenlehre
470 Hoppe, E.. Histoire de la Physique . Paris: Payot,1928. P.96; compare with Kuhn
Sim Disc n.44 p.86.
471 Auerbach, F."Feld, Potential, Arbeit, Energie und Entropie.", in "Grundbegriffe".
In Handbuch der Physik. A.Winkelmann ed.2nd ed. Leipzig: Barth, 1908. 1st vol. Pp.68-91.
472 Cassirer, Ernst. Das Erkenntnisproblem in der Philosophie und Wissenschaft der
Neueren Zeit , Berlin: B.Cassirer. 4th vol tr by William Woglom and Charles Hendel. The
Problem of Knowledge. New Haven: Yale U.P., 1950. Pp.49-50 and P.72. He discussed
Poncelet's Traité des Propriétés Projectives des Figures of 1822.

underlined the acceptance of the French engineering tradition in Germany; his
remarks include a review of German textbooks published before Helmholtz's
works 473 and these remarks are also explicitly quoted in Merz 474. As I am going to
show Helmholtz was in fact aware of this more sophisticated tradition and used
the term "Arbeit" ("travail") in its new technical sense in the first chapter of the
Erhaltung . At variance with what Kuhn asserts, he consciously dropped it in the
second chapter to introduce his "sum of tension forces". This conscious meaning
shift is of great importance in understanding Helmholtz's paper.
As to the second factor it has been pointed out that the emphasis on the
interconversion of the forces of nature is not specific to the period 1830-1850 and
thus cannot be attributed to "the availability of the conversion processes" 475.
The third factor that for Kuhn "triggered" the discovery of energy
conservation is the philosophy of nature, and particularly the German movement
of Naturphilosophie . But Haas' book, not unknown to Kuhn, collects a great
number of 'methaphysical' contributions relevant for energy conservation, starting
from Greek atomism 476. These contributions were all outlining the unity,
uniformity and homogeneity of natural phenomena. Naturphilosophie too is
present in the list, but without a privileged role. Kuhn's strategy to show the
relevance of his third factor is the following: this time stressing vis viva
conservation 477, he (correctly) remarks that its metaphysical aspects were
dropped after 1750 and returned a century later. He claims the influence of
Naturphilosophie in this comeback, but he also admits that "The roots of
Naturphilosophie can, of course be traced back .... to Leibniz" 478. Evidence of an
influence on Helmholtz is supposedely found in a controversial remark of 1882,
where Helmholtz recognizes Kantian influences in the 1847 Erhaltung 479. But to
recognize Kantian roots in the Naturphilosophen is different from asserting that
473 Helm Energetik Pp.14-15.
474 Merz European Vol 2. P. 101.
475 Heimann, Peter. "Conversion of Forces and the Conservation of Energy." In
Centaurus 18 (1974):147-61. P.147 and 159.
476 See the detailed analysis in Haas Entwickl ; particularly chapt 5, Pp.31-35 and
ch.6. Pp.35-45. Kuhn in Sim Disc quotes Haas' book in nn. 39, 74, 75, 79, 82, 92.
477 "Though the technical dynamical conservation theorem has a continuous history
from the early eighteenth century to the present.." Kuhn. Sim Disc p.97
478 Ibid n.77 p.97
479 Helmholtz WA 1 p. 68 .

Kant was a Naturphilosophe 480. The methodological role of Kantian, and even
Leibnizian, influences in Helmholtz's 1847 essay ( as I am going to show:
empirical and transcendental causality, conceptual explanation, regulative use of
the principle of conservation) in my view cannot be confused with ontological
commitments typical of Naturphilosophie.
In my view Kuhn's three factors and in general his distinction between
prerequisites and trigger factors are unsatisfactory and do not lead to convincing
historical results. Three elements seem insufficient to explain the emergence of
Helmholtz's Erhaltung and of the different formulations of "energy" conservation.
Kuhn explicitly asserts 481 not to have discussed the dynamical theory of heat 482
and the impossibility of perpetual motion because he considers them as
prerequisites and not as trigger factors. But both were extremely relevant,
specially in the case of Helmholtz, together with the central force hypothesis, a
main point of debate among scientists accepting the dynamical theory.
Among other short-range influences that were relevant for Helmholtz
researches and are not mentioned in Kuhn's paper I want to stress the wave
theory of heat 483 and the already recalled physico-mathematical tradition dealing
with potential theory and the concept of work as a total differential. A detailed
480 see Williams, Peirce. "Kant, Naturphilosophie and Scientific Method." In
Foundations of Scientific Method: The Nineteenth Century. Ronald Giere and Richard
Westfall (eds). Bloomington: Indiana U.P., 1973. Pp.3-22.
481 Kuhn Sim Disc Pp.101-3
482 For discussions of the role of the dynamical theory in the works of four British
scientists see: Smith, Crosbie. "A New Chart For British Natural Philosophy: The
Development of Energy Physics in the Nineteenth Century." In Hist Sc 16 (1978): 231-79;
and: Moyer, Donald. "Energy, Dynamics, Hidden Machinery: Rankine,Thomson and Tait,
Maxwell". In SHPS 8 (1977): 251-68. They differ on the priority: for Smith is Thomson's,
while for Moyer is Rankine's.
483 Brush showed that between 1830 and 1850 a number of physicists believed in a
wave theory of heat, based on the analogies discovered by Melloni between radiant heat and
light. The influences on the "energy" debates are discussed, including Helmholtz's explicit
references in the Erhaltung . Brush, Stephen. "The Wave Theory of Heat: A Forgotten Stage
in the Transition from the Caloric Theory to Thermodynamics". In BJHS 5 (1970-71): 145-67.
Reprinted in: Brush, Stephen. The Kind of Motion We Call Heat. 2vols. Amsterdam: North
Holland, 1976. Vol.2 pp. 303-25.

historical study of this tradition is still wanting 484, but its relevance for the
"energy" conservation debates is great. Not only in fact in the works of
Poisson 485, Green 486, Hamilton 487, Gauss 488, Jacobi, F.Neumann 489 did it precede
Helmholtz's essay (who in 1847 did not know of Green but quoted Gauss), but
also followed it. Helmholtz's remarks on Jacobi's appreciation of the Erhaltung
for recognising its links with the older tradition are well known 490, but also I want
to stress that Clausius, Weber, Kirchhoff, Riemann, C.Neumann, Sturm kept
talking of "vis viva conservation" well after the "discovery" of "energy"
conservation 491. To formulate a conservation principle they thought sufficient the
484 But see: Auerbach, F. "Potentialtheorie". In Handbuch der Physik. A.Winkelmann
ed.2nd ed. Leipzig: Barth, 1908. 1st vol. Pp. 179-210. Hoppe, E. Histoire de la Physique .
Paris: Payot,1928. Pp.566-74; Kline, Morris. Mathematical Thought From Ancient to
Modern Times Pp.522-31 and 681-87.
485 see Wise quoted above.
486 Green, George: Mathematical Papers of the late George Green . N.M.Ferrers ed.
London: Mac Millan, 1871."Editor's Preface" Pp.VIII-IX; "On the Laws of the Reflection and
Refraction" P.245, 248.
P.264.
487 Hamilton "On a General Method in Dynamics" 1834. Rep in Lindsay Histor
488 Gauss, C.F."On General Propositions Relating to Attractive and Repulsive Forces
Acting in the Inverse Ratio of the Square of the Distance". In Taylor's Scientific Memoirs
Vol.II Part X. Pp.155-7.
489 Neumann, Franz. Die mathematischen Gesetze der Inducirten Elektrischen
Ströme C.Neumann ed. Leipzig: Engelmann, 1889. See: Olesko "Neumann".
Sketch . P.12.
490 Helmholtz WA1 Appendix to the Erhaltung P.74; Helmholtz Autobiographical
491 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen, insbesonder uuber das
Prinzip der Erhaltung der Energie." 1871. In Werke 6 vols. Berlin, 1892-4. Vol.4. Pp. 247-99.
Tr. in Phil Mag 43 (1872): 1-20 and 119-49; Clausius, Rudolf. De la fonction potentielle et du
potentiel. Tr. by F.Folie. Paris: Gauthier-Villars, 1870; Clausius, Rudolf. "Ueber das Verhalten
des elektrodynamischen Grundgesetzes zum Prinzip von der Erhaltung der Energie und über
eine noch weitere Vereinfachung des ersteren." In Pogg Ann 157 (1876):489-94; tr.in Phil
Mag s5 1 (1876): 218-21; Clausius, Rudolf. Die Mechanische Wärmetheorie 2nd ed. 2nd
vol. Braunschweig,1879; Sturm, Jacques. Cours de Mécanique , M.Prohuet ed. Vols. 2. Paris:
Mallet-Bachelier, 1861; Riemann, Bernhard. Schwere, Elektricität und Magnetismus, nach den
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
"energy" was not stressed as an autonomous physical concept but that what was
really supposed to be important was a formal condition on the work done by the
forces. An analysis of this tradition and of its relations with Helmholtz's
Erhaltung and later works ( Helmholtz's attempted in Königsberg at convincing
F.Neumann of energy conservation 492, while was actually F.Neumann who taught
the potential theory to Helmholtz 493) would outline some interesting differences
between mathematical and theoretical approaches to energy conservation.
Once that we recognise that something relevant happened in the history of
conservation ideas in the 40's and 50's of last century more and more compelling
appears that to try to clarify what happened the stress should be put on
differences rather than on similarities.
This leads to reexamine Kuhn's choice of the twelve "pioneers" and the
three groupings. The exclusion from group one of Clausius 494, Rankine 495 and
Kirchhoff: see Hoppe Histoire P.99; Neumann, Carl. Die Gesetze von Ampére und Weber.
Leipzig: Teubner, 1877.
492 Koenigsberger HvH P.64
493 Koenigsberger HvH . Pp.100-1. Helmholtz in his two main papers in
hydrodynamics (Helmholtz, Hermann. "Ueber Integrale der hydrodynamischen Gleichungen,
welche den Wirbelbewegungen entsprechen." In Journ. f. d. reine u. angewandte Mathematik
55 (1858): 25-55; rep. in W A 1 pp.101- 134; "Ueber discontinuirliche
Flüssigkeitsbewegungen." In Berliner Monatsberichte (1868): 215-28; rep. in W A 1 pp.146-
157) and in his electrodynamics works of the seventies drew more on potential theory than on
his 1847 version of "energy" conservation.
494 On Clausius see: Helm Energetik Pp.70-80; Daub, Edward." Atomism and
Thermodynamics." In Isis 58 (1967): 293-303; Clark, Peter. "Atomism versus
Thermodynamics." In Method and Appraisal in the Physical Sciences: The Critical
Background to Modern Science, 1800-1905. Colin Howson ed. Cambridge: Cambridge U.P.,
1976, pp.41-105; Truesdell, Clifford. The Tragicomic History of Thermodynamics. 1822-
1854. New York: Springer,1980; Yagi, Eri."Clausius's Mathematical Method and the
Mechanical Theory of Heat." In HSPS 15 (1984): 177-95; Wolff "Clausius".
495 On Rankine see: Hutchinson, Keith."W.J.M. Rankine and the Rise of
Thermodynamics." In Brit J Hist Sci 14 (1981): 1-26; see also Bevilacqua En and Electr .
Pp.80-90.

W.Thomson 496, is striking. In fact each of these four scientists gave a formal
expression of a "principle of conservation". They all gave substantial
contributions in the period chosen and their relations with Helmholtz's approach
are indeed relevant.
The history of the experimental determination of the equivalent between
work and heat is very sophisticated 497. Kuhn identifies as contributors to this
determination the members of the first two groups, that is: Mayer, Joule, Colding,
Helmholtz; Carnot Seguin, Holtzmann, Hirn. But, as I am going to show,
Helmholtz has to be excluded: we are then left with seven "pioneers". Planck 498
and later almost identically Helm 499, referring to the same period discussed by
Kuhn, cited another seven authors who provided determinations: Clausius,
Kupffer, Favre, LeRoux, Bosscha, Quintus Icilius, Matteucci. They discussed
thirty-one different determinations, including four of Hirn's. Kuhn does not seem
to be aware of these analysis: about Hirn he asserts that "none of the standard
496 On W.Thomson see: Smith, Crosbie. "Natural Philosophy and Thermodynamics:
William Thomson and 'The Dynamical Theory of Heat'." In Brit J Hist Sci 9 (1976): 293-319;
Smith, Crosbie. "William Thomson and the Creation of Thermodynamics, 1840-1855." In Arch
Hist Exact Sci 16 (1976-7): 231-88; Smith, Crosbie. "A New Chart For British Natural
Philosophy: The Development of Energy Physics in the Nineteenth Century." In Hist Sc 16
(1978): 231-79; Wise "Math Route" ; Wise, Norton and Smith, Crosbie. "Measurement";
Wise, Norton and Smith, Crosbie. Energy and Empire: William Thomson, Lord Kelvin, 1824-
1907. Cambridge: Cambridge University Press, 1989.
497 A careful discussion of the different determinations of the equivalents was already
made in1869 by Sacchetti, in 1880 by Rowland, and also in 1906 by Graetz. See: Sacchetti,
Giovanni "Considerazioni intorno all'origine della teoria meccanica del calore." Memorie
dell'Accademia di Bologna VII, 2 (1869): p.149; Rowland, Henry. "On the Mechanical
Equivalent of Heat with Subsidiary Researches on the Variation of the Mercurial from the Air
Thermometer and on Variation of the Specific Heat of Water." In Proc.Amer Acad (2), VII,
(1880): p.75; Rowland, Henry. Relazione Critica sulle varie determinazioni dell'equivalente
meccanico della caloria . Venezia: G.Antonelli, 1882; Graetz, L. "Das Mechanische
Wärmeäquivalent." In Handbuch der Physik. A.Winkelmann ed. 2nd ed. Leipzig: Barth, 1906.
3rd vol., 537-561.
498 Planck Princip chapt. 1 exp.Pp.74-6 and 80-3.
499 Helm Energetik Part1 chapt 7. exp P.34.

histories cites these (Hirn's) articles and even recognizes the existence of Hirn's
claim" 500.
In my view, for a correct grouping of the "pioneers", the acceptance of a
principle of conversion with constant coefficients or of a principle of
conservation of "energy" must be kept separate from the specific model adopted
for "energy", from the experimental determination of the work equivalent and,
finally, from the mathematical representation of the quantity conserved.
Helmholtz in fact thought possible to formulate the principle of conservation in
the Erhaltung with either heat models (caloric and mechanical), and the choice
in favour of the mechanical one was made on experimental grounds. I want also
to underline that the Erhaltung was largely independent from the experimental
determination of the work-heat equivalent: as I am going to show, Helmholtz did
not give much importance to Joule's (wrongly-translated) values. Finally,
mechanical theory is not the only route to the work-heat equivalence: Mayer
denied the first and determined the second.
The need for these distinctions is shown in Carnot's case. Kuhn asserts
that Sadi Carnot's Réflexions "are incompatible with energy conservation" and
"Carnot's version of the conservation hypothesis is scattered through a notebook
written between the publication of his memoir in 1824 and his death in 1832." 501.
Thus he considers Carnot as a pioneer of energy conservation for his
posthumously published acceptance of the mechanical model of heat and of the
relative determination of the work-heat equivalent (370Kgm) 502. But this result,
while obviously placing Carnot among the pioneers of the mechanical theory of
heat and of the determination of the work-heat equivalence, was not needed in
order to consider him among the followers of "energy conservation". Given that
Carnot correctly based his theory on the impossibility of perpetual motion, even
his published results do not contradict the principle of "energy" conservation.
They contradict the mechanical model of heat: Carnot's assumed an interpretation
of "energy" (heat by temperature variation, and this "energy" equals work) that
later was abandoned because the conceptual model of heat adopted (heat as a
substance) was shown to be experimentally wrong. The soundness of Carnot's
500 Kuhn Sim Disc P.68 n.2.
501 Kuhn Sim Disc P.93 and 67 respectively.
502 At the same time Kuhn asserts that, strictly speaking, Holtzmann, who also used
the caloric theory, should not be included in the list. However Kuhn includes him because he
made a determination similar to Mayer's one. Ibid p.67, n.2.

published approach for "energy" conservation is shown by the fact that
Clapeyron's numerical work-equivalent, if reinterpreted in the new energy model,
is not far from later accepted values 503. Thus the notebook is relevant as an
indication of Carnot's shift from one model to another, but not of his acceptance
of the conservation principle.
In my view history would be better clarified if the grouping were done
differently from how Kuhn did it, that is not with the aim of asserting strict
similarities between the members of a group, but with the aim of a more
consistent comparison of their assertions on specific points. I can suggest for
instance to compare the works of the "pioneers" on the basis of:
a) the kind of general principle adopted: a correlation 504 of forces, that is,
a conversion with constant coefficients 505; a conservation at positions (vis viva
principle) 506; a conservation of an underlying unity during a process 507.
503 Clapeyron's coefficient of 1.41Kgm ( work obtained in the passage of a calory
from 1°C to 0°Celsius) is consistent with later results: if we want to shift to the other
interpretation of energy (heat equals work) we have only to multiply it by 273.16 (absolute
temperature of melting ice). We get 385.15 Kgm, a value not too different from Joule's. See:
Planck Princip . Pp.13-4 and 188-9.
504 Kuhn recognizes that a distinction has to be made between correlation and
conservation: Kuhn Sim Disc p.79 and 82.
505 As in Roget, Séguin, Liebig, Colding, Mayer, Hirn, Grove, Faraday, Joule; on
Liebig see: Kremer, Richard. "The Thermodynamics of Life and Experimental Physiology,
1770-1880." Dissertation. Harvard University, 1984. Dissertation Abstracts International, 45
(1985), 3731A. Pp.193-215; on Colding see: Dahl, Per. "Ludwig A. Colding and the
Conservation of Energy." In Centaurus 8 (1963): 174-88; on Mayer see: Lindsay, Robert
Bruce. Julius Robert Mayer, Prophet of Energy. Oxford and New York: Pergamon
Press,1973; Heimann, Peter. "Mayer's Concept of 'Force': the 'Axis' of a New Science of
Physics." In HSPS 7 (1976) : 227-96; on the analogies between heat model in Mayer and Hirn
see: Planck Princip . Pp.75-6; on Grove see: Cantor, Geoffrey. "William Robert Grove, the
Correlation of Forces and the Conservation of Energy." In Centaurus 19 (1976): 273-90; on
Faraday see: Gooding, David. "Metaphysics versus Measurement: the Conversion and
Conservation of Force in Faraday's Physics." In Annals of Science 37 (1980): 1-29. For
remarks on Kuhn and Heimann see n7 and 8 Pp.2-3, n15 P.4; for Grove, Faraday and Joule
see: Heimann "Conversion" pp.148-9; for the differences between the last two, see: Cantor
"Faraday and Joule on Energy Conservation". Paper presented at the Joule' s Centenary
meeting, Manchester 1989. Forthcoming.

) the experimental determinations of the work equivalents.
c) the conceptual model adopted for heat: caloric 508, basic quantity 509, or
mechanical 510.
d) the kind of mathematical formulation adopted: for instance energy as a
sum of two basic forms, positional and kinetic 511 or as a product of an intensity
and a capacity factor 512.
Other interesting comparisons among the pioneers could be done on the
basis of e) the different nationalities 513 and of f) the different generations of the
scientists 514.
506 That is: existence of a potential or of a generalised potential, for instance in
Weber, Clausius, C.Neumann; see my Energy and Electr Pp.75-8; 98-108; 122-36; 153-165.
507 According to Kun: Colding, Helmholtz, Liebig, Mayer, Mohr, Sèguin; see: Kuhn
Sim Disc P.94; but at P.68 Grove and Faraday shared the same idea; Grove is supposed to
announce a universal conservation principle p.70, but compare with Cantor's "Grove" p.286;
also Kuhn's inclusion of Sèguin in this group is not without contradictions: in Kuhn Sim Disc
P.70 Séguin is supposed to have discussed only a special case of energy conservation and at
P.76 and 81 to have ignored the new conversion processes entirely.
508 as in Carnot, Clapeyron, Hess, Holtzmann; the literature on Carnot is huge; see up
to 1984: Home, Roderick. The History of Classical Physics. A Selected, Annotated
Bibliography. New York: Garland, 1984. Pp.238-48; on Clapeyron, Hess and Holtzmann see
Helm Energetik Pp.32-3 and 58-63.
509 As in Mayer
510 As in Mohr, Joule, Helmholtz, Rankine, Clausius, W.Thomson; on Mohr see:
Helm Energetik part 1 chapt 3 P.9; on Joule see: Cardwell, Donald S.L.. "James Prescott
Joule and the Idea of Energy." In Phys.Educ. 24 (1989): 123-7; Cardwell, Donald S.L. James
Joule. A biography. Manchester: Manchester U.P., 1989.
511 For instance in: Helmholtz, Rankine, Thomson.
512 For instance in Rankine.
513 Stress could be given for instance to the differences between British, French and
German institutional and philosophical backgrounds, as suggested by Elkana in: Elkana A case
of? p.56; Elkana, Yehuda. The Discovery of the Conservation of Energy . London:
Hutchinson Educational, 1974.
514 To "highlight differences in background, education, style" as hinted by Cantor
after the example of Caneva. See: Cantor Locating p.9; Caneva, K."From Galvanism to
Electrodynamics: The Transformation of German Physics and its Social Context." HSPS 9
(1978) 63-159.

The widespread lack of discussion in the contemporary secondary
literature of the great works on the history and meaning of the principle of
conservation of energy written at the turn of the century seems an (intentionally)
missed opportunity 515. Detailed analysis of the works of all the "pioneers"
mentioned so far appear in these "classic" works, often the clarification of the
problems at issue is very sharp and the different meanings of principles, models,
mathematical tools, experimental techniques are precisely outlined. This lack of
discussion entails also the problem of the originality of the modern
interpretations. In fact, always taking Kuhn's paper as an "exemplar", it is
possible to trace back some main themes of his analysis: the refusal of the
priority approach to Merz 516 and Haas 517, the relevance of the conversion
processes, of the concern with engines and the related importance of the "work"
concept to Helm 518, Merz 519 and Haas 520; and the "influence" of Naturphilosophie
again to Haas 521.
Given that the mentioned 'classic' works deal with more 'actors' and in
greater detail than what Kuhn does, I am inclined to think that in Kuhn's paper
the selection presented and the way in which it is presented depends on the
particular problem chosen: to explain the origins of the "simultaneous discovery".
An alternative starting point is then needed. How is Helmholtz's paper
framed by other historians? The problem of Kantian influences has been taken up
by Heimann, with an analysis centred on Helmholtz's justification of the central
515 Kuhn in fact quotes Helm, Haas and Mertz, and Planck's 1887 treatise is discussed
by Hiebert in: Hiebert, Erwin. The Conception of Thermodynamics in the Scientific Thought
of Mach and Planck. Freiburg i. Br.: Ernst Mach Institut der Fraunhofer Gesellschaft zur
Forderung der angewandten Forschung, 1968. (E.Mach Inst. Bericht Nr. 5/68). Elkana avoids
discussing Planck's treatise simply asserting that: "his (Planck's 1887) insights and explanations
are far beyond any methodological school." see: Elkana A case of? P.31.
516 Merz, John. A History of European Thought In The Nineteenth Century. 2
vols.Edinburgh and London: Blackwood, 1903. Vol2 p.98.
517 Haas Entwickl . P.98.
518 Helm Energetik Pp.7-34.
519 Merz European vol 2 Pp.100-1 and 104-11.
520 Haas Entwickl.. Pp. 63-92.
521 Ibid Pp.35-45..

force principle 522. Elkana tried to show 523 that Helmholtz was the "true"
discoverer of energy conservation, but centres his attempt on the 1847 paper and
fails to see, among other problems 524, all the later reformulations of the principle.
Cantor points at the difficulties of framing Helmholtz 525, but I think that Winters
offers a good contribution in analysing Helmholtz's Erhaltung in comparison
with his later works on energy 526 . Necessary I believe is also to discuss the use
and modifications of the principle in the debates of the following decades 527.
___________________________________
From such an enlarged point of view, the whole problem of "simultaneity" and of
"discovery" of "energy conservation" should be abandoned. What Helmholtz
really did in his Erhaltung was to lay the foundations of theoretical physics,
through the conscious interplay of conceptual models, regulative principles,
mathematical techniques and experimental results. The whole interplay was
based on an explicit methodology and was meant to have the greatest generality
of possible applications. From then on in physics the theory-experiment interplay
was joined and often overcome by the theory-principle one. From the point of
view of Helmholtz's approach, the other contributions can be seen as less
sophisticated and less conscious, but not less interesting, approaches. At the end
of the century the great debates of the "now mighty theoretical physics" 528 show
the progress achieved: the mechanic, energetic, thermodynamic and
electromagnetic views of nature are the frameworks within which basic studies
on the history of energy conservation were written. I believe that contemporary
history of energy studies can be enhanced if the problem is seen in terms of the
522 Heimann, Peter. "Helmholtz and Kant: The Metaphysical Foundations of 'Über
die Erhaltung der Kraft' ." In SHPS 5 (1974) : 205-38; P.208 n.13, and P.223.
523 Elkana, Yehuda. "The Conservation of Energy: A case of Simultaneous
Discovery?" In Arch Int Hist Sci 24 (1970): 31-60.; "Helmholtz's 'Kraft': An Illustration of
Concepts in Flux". In HSPS 2 (1970): 263-98; The Discovery
524 see: Clark, Peter. "Elkana on Helmholtz and the Conservation of Energy." In
BJPS 27 (1976): 165-176.
525 Cantor Locating
526 Winters, Stephen. "Hermann von Helmholtz's Discovery of Force Conservation."
Dissertation. The John Hopkins University, 1985.
527 see my Energy and Electr Pp.59-74 and 110-21.
528 C.Jungnickel,R.McCormmach, Intellectual Mastery of Nature , 2 vols, Chicago:
Chicago U.P., 1986. 2nd vol.

irth (in 1847) and the development of theoretical physics, as a discipline
different from the experimental and the mathematical.
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