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

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.