The use of the conservation of living force before Helmholtz

ABSTRACT

In his recent authoritative Helmholtz and the Conservation of Energy, Kenneth Caneva has claimed that earlier authors had invoked the principle of conservation of living force only in cases of a system returning to an earlier state, or of one without Newtonian forces. Relaying on texts in the tradition of the French Analytical Mechanics form Lagrange to Coriolis, I argue that this was not the case, and that the principle had been formulated and used for cases where living force proper (mv²) was not conserved but its sum with an integral function (refers today as potential) was constant. In addition. I show that contrary to Caneva’s claim, the principle had been connected to the impossibility of creating power out of nothing. The two points indicate a stronger link between the analytical tradition and Helmholtz and his readers than usually portrayed, and the significant contribution of mechanics to the emergence of energy conservation. On a methodological level the use of living force shows that a common term and even a concept, like energy, is not always needed for its successful employment.

KEYWORDS:

• Energy conservation
• analytical mechanics
• conservation of vis viva
• Joseph Louis Lagrange
• Claude Louis Navier

1. Introduction

The principle of the conservation of vis-viva, and analytical mechanics in more general have occupied a special place in the discussion of the origins of the principle of energy conservation. The formal affinity between the two principles is well known, and early proponent of vis-viva had often suggested a general view of conservation in nature, similar to the one expressed later by energy conservation. Yet, historians, especially from the second half of the twentieth century have correctly emphasized the differences between the two principles. They further tended to diminish the contribution of the vis viva principle and the tradition of its employment to the emergence of energy conservation.¹

In the most formal and comprehensive of the earlier formulations of energy conservation, Hermann Helmholtz presented his new principle of the conservation of force as an extension of the principle of the conservation of living force (vis viva).² The connection drawn by Helmholtz led a few scholars of his work to examine the employment of living forces and their conservation in the analytical mechanics. In his recent Helmholtz and the Conservation of Energy: Contexts of Creation and Reception the most exhaustive, meticulous and authoritative study of Helmholtz’s contribution and its context, and in a more detailed paper on the tradition of living forces in analytical mechanics, Kenneth Caneva examined the approach to the older principle in this tradition in order to asset its role in the emergence of Helmholtz’s thought. Caneva found that Helmholtz was indebted to the tradition in realizing the need for an ontology of massy particle subject to central forces.³ Yet, he indicated a gap between Helmholtz’s thought and that of the analytical tradition, especially from the late eighteenth century on, regarding his belief in a general conservation in nature and the connection that he made between the living force principle and the impossibility of perpetual motion. I agree with the former: scientists in the analytical tradition, whom I discuss below, limited the conservation to particular mechanical systems. Yet, I think that the principle of living force had been invoked in denying perpetual motion, or more exactly in denying the possible production of motion out of nothing. I will discuss that only briefly here since I have elaborated on the employment of proto energetic arguments, which were connected to the principle, in a previous publication.⁴

The main target of this paper is Caneva’s claim that the principle of conservation of vis viva had been stated only for a system returning to an earlier state, or to one without accelerating (Newtonian) forces. Below l show that this is not the case and that the principle had been discussed and more importantly used also for systems under integrable forces, where the value of the living forces was not constant. In other words, it was used also when the vis viva proper (mv², where m is the mass and v the velocity) was not conserved but its sum with twice a function equals to the modern potential (V) (was constant (i.e. $(m{v}^2/2)+V$), as defined for example by Lagrange). Caneva’s claim is based on his distinction between the principle of conservation of living force and the principle of living force, without a mention of conservation. Only the latter, he holds, had been used also for cases were the sum of the living forces proper was not conserved, but unlike the energy principle this was not a conservation principle. Although such an analytical division in possible and Caneva is correct in pointing at some authors who referred to the principle (or law) of living force without mentioning conservation, I doubt that many practitioners in the tradition of rational mechanics recognized the distinction. Caneva does not refer to a contemporary discussion of the distinction, and doubts that Helmholtz made the distinction.⁵ Moreover, I show below that a few central savants used the term conservation and employed the related principle to discuss change of state under forces.⁶

Caneva’s publications will probably remain the definitive study not only of Helmholtz but also of much of the context of the creation and reception of energy conservation for many years. It is therefore important to correct them, in the few places in which they err. Moreover, my analysis indicates further similarities and possible connections between the tradition of analytical mechanics and the related work of Helmholtz and his contemporaries, than suggested by most previous historians including Caneva, and thus indicates the importance of that tradition in the formation of energy physics. In my general claim and in the analysis of analytical mechanics I follow the works of Ivor Grattan-Guinness and Olivier Darrigol who illuminated conceptual developments within French mechanics and their relation to the later emergence of energy conservation.⁷ The question at stake, namely how did mechanists regarded the conservation of living force, has also interesting bearing on the way scientists exploit potentially useful relations in their mathematical formulation and the power of the formulation, also when concepts seem to lack either coherence or completeness, as was the case with the conservation of living force.⁸ I show below that clear terms were not always needed for successful employment of mathematical formulations to concrete systems, like that of the sun and planets.

2. Lagrange

Although the principle of conservation of living force was well known much earlier it is convenient to begin my discussion with Joseph Louis Lagrange and his influential book Analytical Mechanics of 1788. There, he deduced the equation (1) $\sum m\left({\displaystyle \frac{d{x}^2+d{y}^2+d{z}^2}{2d{t}^2}+\mathrm{\Pi }}\right)=F$(1) where the sum in over all the bodies of a system, x, y, z, t are the coordinates of a body, m is its mass, Π is the path integral of the forces per mass impressed on the body (each force by its own displacement), assuming that the scalar product of force by infinitesimal displacement (in modern terms) ‘is integrable, [which] is always the case with accelerating forces directed at fixed centres, or with bodies of the same system, that are proportional to some functions of the distance’; and F is an arbitrary constant of integration.

This last equation [he explained] expresses [renferme] the principle known by the name of the conservation of living forces. . . one sees from this equation that [the] living force is equal to the quantity $2F-2\sum m\mathrm{\Pi }$, which depends only on the accelerating forces acting on the bodies.⁹

Thus, for Lagrange the principle expressed not the conservation of the living force proper (either in singular or in plural) but of its sum with the unnamed path integral on the forces F. In later places in the book he employed symbols better known for the modern reader: T for the living force divided by two and V for $\mathrm{m}\mathrm{\Pi }$ and wrote ‘ $T+V=\mathrm{const}$, which expresses the conservation of living forces’.¹⁰ This suggests the employment of the principle for calculating the living force under changing conditions of the force function V. Interestingly, in the later references to the equation Lagrange did not use a symbol (F) for the sum, as he did in introducing the conservation of the living force.

Following the common usage, Lagrange referred to the principle of conservation of living forces, although he did not mention specifically cases where the living forces proper are conserved. On the contrary, as quoted above, he suggested a way to use the principle to calculate the change in the value of the living force of a body.¹¹ Although he called it a principle, Lagrange deduced the law and its related equations (like (1)) from what he considered as the basic laws of mechanics. It was not a fundamental principle of his system as the conservation of energy would become later. Moreover, it was valid only for the cases of integrable forces, like central forces, which Lagrange mentioned. It was not valid to other systems, for examples those subject to friction, which is the common case in terrestrial phenomena. Lagrange’s followers in the analytical tradition continued to restrict the validity of the conservation to particular systems.

In his Théorie des fonctions analytiques of 1797 Lagrange made the relation between the living forces and the product of the acting (Newtonian) forces by the path travelled more explicit. One way was by writing the living force as equal to the initial living force minus the difference in the functions on the forces: $M{u}^2+N{v}^2+\mathrm{\&}\mathrm{c}.=\mathrm{M}{U}^2+N{V}^2+\mathrm{\&}\mathrm{c}.+2F(p,q...)-2F(a,b...)$where u, v are the velocities and p, q the coordinates at a particular instant, U, V and a, b the initial velocities and coordinates at, and M and N masses. ‘This equation expresses the principle of the conservation of the living forces taken in its full generality’. Collecting the initial and terminal living forces in one side and the integral on the forces in the other he further wrote: $M{u}^2+N{v}^2+\mathrm{etc}.-\mathrm{M}{U}^2-N{V}^2+\mathrm{\&}\mathrm{c}.=2F(p,q...)-2F(a,b...)$Such formulation dispenses with the constant of integration (F in eq. 1) and refers directly to relations. It is useless for cases of return to the same initial positon as it becomes trivially zero on both sides. Lagrange further discussed the path integral function on the forces showing that it equals to the product of the force by the path the body travelled when the force is constant. He further concluded:

Thus, when these [accelerating] forces act upon bodies connected to each other in any manner, they produce, in the entire system, an increase in living force equal to the sum of the living forces that each force would produce, individually, if it acted alone on a free body, and that it made the body travel along its direction, a distance [espace] equal to that that the body really travels along the same direction. This is strictly [proprement] what constitutes the principle of conservation of living forces.¹²

Thus, for Lagrange the principle of conservation of the living forces stated the ability to calculate their magnitude from the integral function on the accelerating forces, by separating the effect of each force on each body. It was not confined to cases were the living forces proper kept their value. He saw in the equation that expresses the conservation a way to calculate the gain or loss of living forces, or the force function. A return to the same conditions was superfluous.

Lagrange provided a similar discussion on the relation between the living force and the action of the forces in the 1811 second edition of Mécanique analytique. The later one is closely similar to d’Alembert’s formulation in his 1743 Traité de dynamique (and in the 1758 second edition) suggesting its influence. Yet the differences between the two mechanists are telling. D’Alembert referred to ‘two principles that consists what one calls the conservation of living forces’.¹³ One was for the case without the exertion of force, and the other for the case with forces, for which he suggested a way to calculate the change in the living force. Lagrange, on the hand, referred to one principle of conservation, which was strictly about cases in which accelerating forces are involved. D’Alembert’s statement is the closest that I found to the distinction between two principle of living forces claimed by Caneva, yet none of them is about a system returning to an initial state. In this context, it should be mentioned that Lagrange sometimes omitted the term conservation (or the term principle) in discussing the principle or equation of living forces, without apparent change of meaning. It seems to be only a manner of abbreviation.¹⁴

A second difference concerns the centrality of the principle. While d’Alembert introduced it in the last chapter of his treatise, the conservation of living forces was a basis for many further results in Lagrange mechanics. In concluding his discussion of living forces in fonctions analytiques, Lagrange indicated a connection between their principle and the theory of machines. The sum of the living forces and the integral on the forces is not conserved in such cases. Yet, the equation of its conservation provided Lagrange a way to assess the loss to resistance in machines, by comparison to a case of conservation, where there is no resistance. He exemplified that with a particular calculation for the case of a constant force – the living force of a weight P descending a height h which is 2Ph. In this uncharacteristic detour into real machines, Lagrange was probably influenced by the emerging engineering tradition of the late eighteenth century with its discussion of real machine and the loss of living force.¹⁵ The connection to real machines, which are not conservative systems continued to be important in the latter tradition.

In the second edition of mécanique analytique Lagrange extended the role of the principle. Now he kept a consistent sign – H to mark the conserved sum of the living force and the displacement integral on the force. He repeatedly mentioned its constancy, and referred to it along the book and to a short version of the equation of the conservation of the living force as: $T+V=H$, which appeared without H in the first edition. Still, he neither suggested a name for this constant nor had a unified concept for it.¹⁶ The lack of a conceptual understanding of H did not prevent its use. He showed that the value of H indicates the shape of an orbit of a celestial body revolving around a star (negative value indicates an elliptic orbit, zero a parabolic and positive a hyperbolic). This conclusion was part of his discussion of a system of bodies under central forces, based on the equation of the conservation of living forces. Lagrange mentioned that the equation does not hold when the medium resists to the motion. But since in space there is no resistance, he deduced the equations of motion of point masses under a central force, which is the case of the planets and comets around the sun, deriving, among others, equations for the location of such a body in time. On these equations, he elaborated further equations that characterize the motion of the heavenly bodies regarding eccentricity, apsides, etc.¹⁷ Lagrange often used the equation of conservation of living forces and the constant H in cases of action under forces, usually without mentioning its name.¹⁸ His attentive readers probably identified such cases as the application of the conservation of the living forces. Thus, in words and practice Lagrange claimed and showed that the equation of the conservation of living force is valid and useful in systems under accelerating forces, i.e. when the value of the living force proper is not conserved.

3. Poisson

In his 1811 Traité de mécanique Siméon-Denis Poisson asserted that the magnitude of the living forces proper is conserved only in returning to the same conditions. Caneva quotes the following:

The increase or diminution of the living forces, on passing from one position to another, will depend not at all on the curves described by the bodies; this increase will be zero, and the sum of the living forces will again become the same, every time the system will return to the same position; finally, this sum will be conserved constantly the same when the bodies are not acted upon by any accelerative force.¹⁹

These were three common conclusions from the equation or principle of living forces, i.e. path independence, conservation of the value of living force in returning to the initial position and its conservation when no forces are involved. They reappear also in the writing of later authors. Note, that the path independence indicates that the principle and the equation of living forces are applicable to bodies subject to forces also when the system does not return to a previous state. That became clear in the sentence just before this quotation. In a system in which the equation is valid

the sum of the living forces ∑mv², at any time is given by the equation $[\sum \mathrm{m}{\mathrm{v}}^2=\mathrm{C}+2\phi (\mathrm{x},\mathrm{y},\mathrm{z},{\mathrm{x}}^{\mathrm{\prime }},{\mathrm{y}}^{\mathrm{\prime }},{\mathrm{z}}^{\mathrm{\prime }},\mathrm{etc}.)$], when one knows the value of this sum at a particular moment and the coordinates of the bodies in these two positions of the system.

Thus, in similar to Lagrange’s statement, Poisson assumed that the equation could be used to calculate a change in the magnitude of the living force.

Moreover, Poisson saw the ability to calculate the living forces as a major property of their principle of conservation. Following the four conclusions from the equation, just quoted, Possion wrote:

One sees by this [equation], how the theorem regarding the velocity square that we have found [v² − A² = 2f (x, y, z) − 2f (a, b, c)] in considering the motion of an isolated material point, extends to motion of a system of bodies. This theorem is known under the name the general principle of the conservation of living forces. [f is the path integral on the forces, v and A initial and terminal velocities]²⁰

Caneva remarked that Possion’s last sentence is ‘odd’.²¹ But this is only because the historian assumed that the conservation of living forces did not include cases of changing the value of the living forces proper. Apparently, this was not the view of the French Savant. In his formulation of the principle, Poisson followed Lagrange rather than his mentor Pierre-Simon Laplace. The author of Mécanique céleste restricted the conservation of living forces to systems where their value proper remains constant.²² Likely convinced from its usefulness, Poisson prefer an extended version of the principle.

Thus, for Poisson the conservation of living forces was a useful theorem for calculating changes under the integrable forces. Like Lagrange in fonctions analytique of 1797, he suggested an equation of differences, which manifests the balance between the living forces and the unnamed integral function on the forces. The equation was only implicit as he did not write the general equation for many bodies, and when he wrote the equation for one body he did not mention the living forces. He made the equation explicit in the second edition of 1833.²³ Yet, like all others in this tradition Poisson referred explicitly to the limits of validity of the principle, deducing, for example ‘Carnot’s theorem’ that expresses the loss of living force, due to burst changes of velocity in machines.

Poisson not only stated the usefulness of the equation of conservation but also employed it to gain particular results, for example to analyse oscillations of a floating body disturbed from equilibrium position, examining whether this was a stable equilibrium. He elaborates an expression for the magnitude of the path integral of the gravitational force – $\phi$. The value of $\phi$ depends on the relation between the positions of the centre of mass of the whole body and the centre of mass of its part below the surface. In this case the value of the integral $\phi$ provides a way to know whether the body would be in stable equilibrium or not. Since the living force of the body when it is released from a position of a disturbed equilibrium is zero (like the velocity), $\phi \mathrm{equals}$ the sum of the living forces and the integral function. This makes it similar to Lagrange’s H, which was also defined by the value of the living force at a particular moment (see fn. 16). Poisson’s treatment resembles Lagrange’s discussion on the kind of motion under different values of H (which appeared in the same year), but here the value of the function was calculated from the parameters of the problem, rather than given. Although he did not mention the term ‘conservation’ in this analysis, he noted that it is based on the principle of living forces and directed the readers to the discussion of their conservation from which I quoted above. The term conservation of the living forces appeared also in some subchapters titles and in the more detailed table of contents.²⁴ Thus, Poisson did not limit the principle of conservation to the return to a previous position.

In the 1833 second edition of the book Poisson suggested a similar and extended discussion of the equation and principle and their consequences but omitted the term conservation, referring to the principle/equation of living forces. He also employed its equation in a few further cases. The omission seems deliberate, but Poisson did not explain why he had changed the name. A plausible reason can be found in his exposure to the developments in machine theory. He appended to this edition an ‘addition regarding the usage of the principle of living forces in the calculation of machine in motion’ of 35 pages based on the works of Navier, Coriolis, Poncelt and their colleagues at the engineering schools. I describe their main ideas below; here it is suffice to mention that the engineering school focused on the loss of living forces in machines. In following this tradition, Poisson continued using the equation and principle of living forces and kept their name, but now they expressed the amount of living force transmitted to useful motion and the amount lost to resistance. Thus, in the ‘addition’ the principle was no longer applied mainly to conservative systems where the sum of the combined magnitude of living forces and the path integral on the forces is constant, but also to system where the sum was not conserved. It was still useful also for discussing such cases but the term ‘conservation’ seemed inappropriate for them. Regarding the particular claim of Caneva, it is worth noting that Poisson kept the same concept of the principle of living forces either with the term conservation (in the first addition) or without it (in the second). He did not distinguish between two different principles.²⁵

4. Navier

In his 1819 notes on and long additions to Belidor’s old book on hydraulic, Claude L. Navier suggested an extensive treatment of the conservation of living forces in analysis of bodies subject to (accelerating) forces. Like Lagrange in 1797 he expressed ‘the principle known as the conservation of the living forces’ as an equation of differences. In a somewhat more elegant and explicit way than his predecessors he wrote (for the case of one body later extended to many): $m{v}^2-mv{{}^{\mathrm{\prime }}}^2=2\int (m\xi dx+m\chi dy+m\zeta dz).$where v’ and v are the initial and terminal velocities, and ξ, χ and ζ the ‘forces’ (more exactly forces per mass) in the three coordinates$.$ mv², he explained, is the living force of the body yet ‘[n]o metaphysical notion should be attached to this expression, but it should be considered merely as a conventional name’. More interestingly, Navier named the integral on the right side: ‘This product of a pressure affecting a body multiplied by the space that this body travelled in the direction of the pressure will be called the quantity of action exerting by the force and impressing the body’. Thereby he adopted the name given by Coulomb and earlier (though not consistently) Euler. Smeaton and Lazare Carnot had given the quantity other names;²⁶ yet in the analytical tradition it was usually nameless, as Caneva has emphasized. For Navier the principle of conservation of living forces indicated that the quantitative change in the living forces equals to (twice) the change in the quantity of action.

The living force acquired during a certain time by a body moved by the action of some forces is always numerically equal to twice the quantities of action that these forces impressed on it during that time, taking the quantities of action in the negative when the spaces traveled are in opposite direction to the action of the forces.²⁷

This principle of conservation is clearly not restricted to a return of a system to an initial position. Navier acknowledged also the consequence of path independence, but did not mention the case where no forces are involved.

Navier explicitly employed the conservation principle in many cases as a way to analyse exchanges between the living forces to the quantity of action. His first simple and well-known example was of a point mass under the earth’s force of gravity, where the conservation of living force allows calculating the velocity of the falling body. More complicated examples led him to analyse pendulum under gravitation, where he deduced the well-known dependence of its period on its length and the gravitation constant. After extending the principle of living forces to systems of many bodies, he resorted to it in discussing also solid bodies and fluids. He examined the rotation of a heavy body around a horizontal axis, developing an equation for the motion of the body’s centre of gravity and finding its centre of rotation. The principle also served him in calculating the velocity of water leaving an orifice, under the force of gravity. In these examples, he equated the changes in the living forces to changes in the quantity of action of terrestrial gravitation, which is the only force active in the hydraulic cases examined in the book.²⁸

Darrigol has concluded that Navier made the quantity of action the main quantity of dynamical interest. Yet, one should remember that it was coupled with the living force, which Navier stated ‘is of the same order and nature’ and with the principle of its conservation.²⁹ Although he suggested neither a common name nor a common symbol for the two quantities, that they were of the same nature made the balance between them close to their common conservation. Clearly for Navier, the magnitude of each of them alone was not conserved but was quantitively linked to the other, so the magnitude of the one can be changed only if the other is changed in the same magnitude. That they were of the same nature made the situation similar to the established conservation of the electric charge (or fluid), where the amount of negative charge cannot be increase without a common increase in the positive charge.

Still, like all other authors in the tradition of rational mechanics, Navier limited these kind of conservation processes to special systems and discussed also the loss of the living force rather than its reversible transformation to the quantity of action. The loss happens with sudden changes of velocities as shown by Carnot’s theorem, and in machines due to resistance that ‘consumes’ quantity of activity (resistance is necessary in real machines).³⁰ The increase in the living force produce by the machine, he explained, equals to the difference between the quantity of action of the applied force and the quantity of action of the resistance. The maximal living force of the water-motor is reached when the process ceases and thus, the excess quantity of action of the motor is in equilibrium with that of the resistance. In this argument, he employed results from his discussion of the principle of conservation of the living force, although he discussed here the case of lost quantity of action. A report for the French Académie on Navier’s work praised his machine theory ‘based on the employment of the principle of conservation of living forces’.³¹

5. The engineering analytical tradition

Navier added an analytical discussion of mechanics to an old engineering book, giving the conservation of the living forces a central role. He elaborated on its relation to machine theory in an ‘addition on the principles of calculating and the installation [établissement] of machines and motors’.³² These devices that harness water power were a central interest in the scientific – engineering analytical French tradition to which he belonged. The quantity of action expressed well the maximal living force that a machine could provide. Navier agreed with Montgolifer that ‘the living force is what we pay for’. As the origins of his terms suggests, he contributed to an existing discourse. His writing and teaching contributed to make it more vibrant in the next decade. In his courses on machine theory during the 1820s Coriolis coined Navier’s ‘quantity of action’ ‘work’, expressing its close relationship to the work done by machines. In his 1829 book, he introduced two connected and more important innovations. (a) He defined the living force as half mv², instead of twice that magnitude, which, he claimed, caused one to disregard the work concept, which he regarded as the more fundamental one. (b) He employed a new name ‘the principle of transmission of work’ to the old principle of living force. He continued using also the name the equation of living forces, which he wrote as a difference equation in similar to Navier, but divided by two.³³

As was often noted, the new terms put more emphasis on the loss of living force and work, than the older ones. The work expended in a process, in general, is equal to or smaller than the gained living force. That was expressed also by Carnot, Lagrange and Navier, inter alia, yet the move to transmission, rather than conservation, made this fact clearer. The term work indicates the connection to actual machines and their non-conservative systems. In the first edition of his book, Coriolis referred to the principle of the transmission of work rather than to the principle of living forces, and avoided discussion of conservation. Still, he employed his new principle in similar to the way Navier employed the principle of conservation of living forces in 1819. In the second extended edition of 1844, he employed more regularly also the terms principle and equation of living forces, but avoided writing about their conservation.³⁴ Others in the engineering tradition followed Coriolis in viewing work as a more basic concept than living force. For example, in his textbook on industrial mechanic, also form 1829, Poncelet stated his preference to the principle of the transmission of work on the principle of living forces, and kept referring to the former within the text without mentioning the principle or the equation of living forces.³⁵

6. Pontécoulant

My last example in also from 1829, but form celestial mechanics: Gustave de Pontécoulant Analytical Theory of the System of the World. Like Poisson, and probably following his mécanique, he inferred the three common corollaries from the equation of living force, which he wrote exactly like Poisson as: $\sum m{v}^2=c+2\phi (x,y,z,x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }},etc.)$these where: the conservation of the living forces without forces, the path independence of its increase and that it is zero and the total living force becomes the same when the system returns to the same position. ‘This theorem, Pontécoulant added, constitutes the law of motion called the principle of the conservation of the living forces’.³⁶ He kept referring to the term ‘conservation’ in invoking the principle in a more elementary treatise of 1840.³⁷

Following Lagrange, he expressed the constancy of the sum of living force and the function on the force (with an opposite sign from the one given to it by Lagrange) as a common constant: $T-V=h$. He employed the equation in different forms as a basis for further elaborations, for cases of motion under the action of central forces, which is the general case in celestial mechanics. At least for the case of the motion of one body under the influence of many bodies (like the sun system) he wrote explicitly that the equation ‘expresses the conservation of the living forces’. According to the equation, the sum of the living forces due to the absolute and relative velocities of the bodies minus twice the sum of the integrals on the forces between the bodies (our potentials with an opposite sign) equals h. Among others, he employed the living forces equation to analyse perturbation forces on a planet, using among others the variable h. He concluded that if these forces are periodic (which should be approximately the case in a planetary system like that of the sun) they do not lead to a secular change in the velocity of rotation and position of the axes of rotation of the celestial body in its orbit.³⁸

7. The conservation of living forces and the denial of perpetual motion

Caneva is right in claiming that

neither the impossibility of creating motive power indefinitely without some compensating expenditure nor the impossibility of perpetual motion was, in Helmholtz’s day and before, treated as a regular concomitant of the principle of the conservation of vis viva in rational mechanics.

Yet, although not usually mentioned in the treatises on rational mechanics, Helmholtz was not the only one ‘who made that connection’.³⁹ Apparently, in the nineteeth century there was no need to deny the possibility of perpetual motion in mechanics itself (but, as mentioned below, Lazare Carnot raised it in the late eighteenth century). Yet, André-Marie Ampère and Sadi Carnot alluded and referred to the conservation of living forces to deny endless creation of power in the realms of electromagnetism and of mechanical power of heat.

Ampère employed the conservation of living forces along with a general notion of the impossibility of creating motion out of noting in denying the possibility of explaining phenomena of electromagnetic rotation by central forces between static poles. He first used it in 1822, without invoking the formal equation. Two years later, he relied on

a rigorously demonstrated theorem … that when the elementary forces depend only on the distances between the material points that exert them, [then] the material points of a system that these forces put in motion cannot all return to the same situation with velocities higher than they had on leaving.⁴⁰

This was in contrast to the observation of a rotation of a current-carrying wire a round a magnet where a motion is created and continues against friction. Thus, he concluded, the rotation originated in motion of electric charges (or fluid) and not by static forces. In his more synthetic and mathematic 1826 book he repeated the same argument, now referring to ‘the principle of the conservation of living forces’, explicating the connection between that principle and the impossibility of creating motion out of nothing.⁴¹

Sadi Carnot alluded to the conservation of living forces without mentioning it by name in his (now) famous 1824 Reflexions on the Motive Power of Fire where he deduced ‘the second law of thermodynamics’. He described a process in which one could have gained an excess of ‘motive power’ (his term for Navier’s quantity of action and our work) in a reversible engine. If that was possible, he explained,

[w]e should then have a case not only of perpetual motion but of motive power being created in unlimited quantities without the consumption of caloric or of any other agent. Creation of this kind completely contradicts prevailing ideas, the laws of mechanics and sound physics; it is inadmissible.

It should have been clear to his contemporaries that the contradiction to ‘the laws of mechanics’ is expressed by the conservation of living forces.⁴² In a footnote, He further connected these mechanical laws to the denial of perpetual motion:

It may be objected at this point that perpetual motion, which has been shown to be impossible for purely mechanical processes, may not be so where the effects of heat or electricity are concerned. But is it conceivable that the phenomena of heat or electricity are due to anything but some kind of motion and hence that they are not governed by the general laws of mechanics?⁴³

The conservation of living forces was the result of the laws of mechanics that showed the impossibility of perpetual motion. The relation between the two was well known. In particular it appeared in the works of Lazare Carnot, Sadi’s father. In his 1783 treatise on machine theory, Carnot the elder showed that perpetual motion is inadmissible since it contradicts the equation of living forces (the sum of the living forces and the ‘moment of activity’, his name for Sadi’s ‘motive power’). Although he discussed their loss (e.g. by abrupt changes of velocity) he referred to the equation as expressing ‘the famous principle of living forces’. Moreover, Lazare claimed that it is ‘this conservation of living forces, which served. . . as a basis for our whole machine theory’.⁴⁴ Students of the son’s work have concluded that

[t]here is much . . . in the reflexions that suggest Sadi’s indebtedness to his father: for example his comments on perpetual motion, his emphasis on the need to consider a closed cycle of operations, and, perhaps most important of all, the notion of reversibility.⁴⁵

Thus, the denial of perpetual motion was connected in the Carnots’ thought and writings to the conservation of living forces.

Thus, this well-known quote of Sadi Carnot show that, like Helmholtz later, he saw in the conservation of living forces a demonstration of the impossibility of perpetual motion. Further, it points at his view, again resembling Helmholtz’s, that the principle should be extended to heat and electricity, and presumably to all physics. In the note, he raised two more arguments, an empirical and a philosophical, for the general impossibility of perpetual motion. In unpublished notes, that were probably written at the time of writing his published treatise or shortly after, Carnot adopted the mechanical reduction of physics.

Hence [he inferred] we may state, as a general proposition, that . . . strictly speaking, motive power is neither produced nor destroyed. . . This principle can be deduced directly, so to speak, form the principles of mechanics. For it follows logically that there can never be a loss of living force or what amounts to the same thing of motive power, if bodies act on one another . . . without true impact. [i.e. without hard collision]⁴⁶

Again, even if not mentioned by name, the allusion to the conservation of living forces is clear, as well as the similarity to Helmholtz’s later extension of the principle of conservation of living force from mechanics to all physics. Unlike Helmholtz, Carnot still needed to claim against the possibility of hard collisions as they entailed a loss of living force. Helmholtz seemed to take their impossibility for granted.⁴⁷ To conclude, both Ampère and Carnot connected the conservation of living force to the denial of perpetual motion, and both extended its realm of applicability beyond mechanics proper, by assuming that other realms are subject to the mechanic-like laws.

8. Conclusions

At least a few important figures in the analytical tradition employed the principle of the conservation of living forces for the sum of the living forces and the path integral on the force, for cases in which the decrease in the value of one of them led to an equal increase in the value of the other. In other words, they used it in a way similar to the later employment of the conservation of energy for calculating transfer of energy from one kind to another, if only for particular mechanical systems in which the former principle was regarded valid, i.e. subject to integrable forces. In practice that was mainly for systems subject to terrestrial and inter-planetary gravitational forces (e.g. pendula, hydraulics and celestial mechanics).

Lagrange, Poisson (in the first edition of his Mechanics), Navier and Pontécoulant employed the term ‘conservation’ also for such cases where the sum of living forces and the force path integral was constant but not each alone. The sources that I have examined do not suggest that contemporaries distinguished between two principles (of ‘the conservation of living force’ and of ‘the living force’ without ‘conservation’) but suggest different views of one principle and some alterations in its name. I know of no example in which two distinct principles were employed or referred to in the same text after d’Alembert. Omissions of the term ‘conservation’ within texts in which it appeared at other places were common, but seem as abbreviations since they did not suggest different meanings, and often the term returned in a later place.⁴⁸ In the one case where the same author avoided the term ‘conservation’ in a latter edition (that of Poisson), his wording changed neither the way he described nor the way that he employed the principle. Authors began dropping the term conservation only in the late 1820s. This was mostly in the engineering school, which influenced also Poisson.⁴⁹ I assume that members of this school regarded the term ‘conservation’ unsuitable not because it referred to the balance between living force and work (rather than to the conservation of the former alone), but since they were interested in non-conservative systems in which work was consumed by resistance.

The examples given here shows a continuity between the way the conservation of living force was employed within the tradition of analytical mechanics and the way Helmholtz later used it. The comparison also highlights the main difference between the two concepts, namely the limited validity of the principle to particular mechanical systems within the analytical tradition, versus the general validity of Helmholtz’s ‘principle of conservation of force’. Helmholtz was correct in viewing his principle as an extension of the old mechanical one through his assumption that all forces are conservative mechanical ones (and his oblivious denial of hard collisions),⁵⁰ to which one should add his focus on the generally conserved quantity – the sum of the living force and his ‘tensional force’. Caneva has shown that many of Helmholtz’s contemporaries failed to see the novelty of his work due to its close connection to the mechanical tradition.⁵¹ The similarity between his employment of the generalized principle and earlier employments of the mechanical principle of conservation makes their reaction even more understandable. It also suggests the debt of Helmholtz and his contemporaries to the tradition of analytical mechanics.

Caneva concluded that

two common aspects of especially later treatments of the conservation of vis viva were essential to Helmholtz’s conceptualization: a mechanics-friendly ontology of massy particles subject to attractive and repulsive forces that depend only on distance, and the notion of the return of a system of particles to an earlier configuration whereby it again possesses the same earlier quantity of vis viva.⁵²

My examples show that such treatments included also discussions of cases where the system does not return to the same positon. These provided explicit references to the constancy of the sum of the living force and (twice) the spatial integral function on the force, and to the transfer from the one to other, with examples of their useful employment. Helmholtz’s claim that ‘[t]he sum of the existing living and tensional forces [the spatial integral on the force] is … always constant’⁵³ follows from his ontology of central forces and the employment of the conservation of living forces (not from its formulation) within rational mechanics.⁵⁴ These contributions of the mechanical analytical tradition to Helmholtz and his contemporary readers indicate that it was more significant to the emergence of energy conservation than commonly claimed in the literature. This is not to deny the importance of other factors like the concern with engines, the source of animal heat (connected to the controversy regarding the existence of ‘vital force’) and the rise of the mechanical theory of heat.

Moreover, although limited to particular systems, the mechanical principle sufficed to show that the sum of living force and the force integral cannot increase, and thus indicated the impossibility of creating ‘motive power’ out of nothing. Ampère and Carnot made that explicit also beyond mechanics, and a central tool in their reasoning. In a previous article, I have shown the significance of their arguments coupled with other pre-energetic notions for the development of energetic physics among other directly and indirectly (through Faraday and Liebig) on Helmholtz. Grattan-Guinness and Darrigol have earlier indicated the importance of inner developments within mechanics, including the concept of work (even if it was not central for Helmholtz) and the denial of hard collisions, which allowed the extension of the conservation of living force to whole mechanics. Coupled with other considerations in the hands of the likes of Saint Venant the latter led to general view of conservation in nature.⁵⁵

From the modern view of energy, it seems strange that analytical mechanics lacked a common name or even a common concept for the sum of the living forces and the integral function (itself usually nameless). Yet, they were not needed in order to acknowledge, announce and successfully employ the conservation of this sum, regarded as a constant. What scientists in this tradition needed were exact definitions of the two magnitudes and a proof of the equation of living forces. It is interesting to note that Helmholtz also failed to defined such a unified concept (‘a concept of force-as-energy’, in Caneva’s terms), which did not hinder him from recognizing the conservation of the sum.⁵⁶ At least within the analytical tradition, the lack of a common name was perhaps connected to a rejection of any metaphysical meaning for the magnitudes involved (a point made explicitly by Navier).⁵⁷ That was no obstacle. Perhaps it even helped the employment of the useful equation of conservation of living force within mechanics, often used without repeating its name, since it freed the mechanist from discussing the physical nature of the involved magnitudes. For analysis, useful equations are sometimes more important than physical concepts. A common symbol for the sum seemed useful for some problems as shown in the works of Lagrange, Poisson and Pontécoulant.⁵⁸

The last point I would like to make concerns the need to examine not only the formulations of physical laws but also of their implementation in practice. For the historians, the way a scientist employed a principle shows how the author understood it. For contemporary and later scientists, less interested in understanding their colleagues and more in solving particular physical problems, the employments provided potentially useful examples to implement in their own writings. Like the historian (and usually more thoroughly than him) they learnt the meaning of the laws from examples, even if their meaning differed from the explicit view of the author. From the practice with the equation of the conservation of living forces it was clear that it is useful for systems subject to forces under changing conditions. The conserved magnitude was the sum of the living forces and the path integral on the force (the modern potential). Even if the terms conservation and principle were often omitted, that they were related to the same equation in another part of the work was enough for the readers to connect it with conservation and transmission between living force and the force function, two central properties of energy conservation.

Acknowledgements

The workshop ‘Conceptual Innovation in Classical Mechanics' (Seville, September 2022) has encouraged me to conduct the research presented here (and earlier there). I thank María de Paz and José Ferreirós for organizing the meeting, the participants for the discussions and the journal’s anonymous referees for their useful comments. As usual I am grateful for the hospitality and library services of the Max Planck Institute for the History of Science in Berlin.

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Notes

1 This is very clear in Thomas S. Kuhn, ‘Energy Conservation as an Example of Simultaneous Discovery’, in The Essential Tension: Selected Studies in Scientific Tradition and Change (University of Chicago Press, 1977), 66–104; and appears also in other comprehensive works like Crosbie Smith, The Science of Energy: A Cultural History of Energy Physics in Victorian Britain (University of Chicago Press, 1998).

2 Since most of my sources referred to ‘living force’ in their own language (French) rather than in Latin, I prefer using its English rather than its Latin term. This also makes clearer Helmholtz’s extension of the principle from living force to force in general.

3 Kenneth L. Caneva, Helmholtz and the Conservation of Energy: Contexts of Creation and Reception (Cambridge, MA: MIT Press, 2021); Kenneth L. Caneva, ‘Helmholtz, the Conservation of Force and the Conservation of vis viva’, Annals of Science, 76 (2019), 17–57, e.g. 55. Since the paper is more detailed than the book I will refer mostly to it here as ‘Conservation of vis viva’. Hermann Helmholtz, Über die Erhaltung der Kraft (Berlin: Reimer, 1847).

4 Shaul Katzir, ‘Employment Before Formulation: Uses of Proto-Energetic Arguments’, Historical Studies in the Natural Sciences, 49 (2019), 1–40.

5 Caneva, Helmholtz (n. 3), 73, ‘conservation of vis viva’ (n.3), 56.

6 Caneva mentions most of the example that I give below. I assemble relevant cases for my claim together and suggest a somewhat different interpretation and context to some of them. I also include here a discussion of the use of the conservation of living forces, which Caneva does not discuss, since it seems to me helpful for understanding its significance and historical role.

7 Ivor Grattan-Guinness, Convolution in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics (Basel: Birkhäuser, 1990), especillay ch. 16, pp. 1046–1121; Ivor Grattan-Guinness, ‘Work for the Workers: Advances in Engineering Mechanics and Instruction in France, 1800–1830’, Annals of Science, 41 (1984), 1–33; Olivier Darrigol, ‘God, Waterwheels, and Molecules: Saint-Venant’s Anticipation of Energy Conservation’, Historical Studies in the Physical and Biological Sciences, 31.2 (2001), 285–353.

8 In this case, one could either employ the concept of living force to include something which is not living force (V in Lagrange’s terms above) and lose coherence or employ it only for the living force proper (mv²) and lose completeness. See also the discussion below.

9 Joseph-Louis Lagrange, Méchanique analitique, 1788, vol. 1: 207–8, emphasis in the original. In the context of such equations the verb ‘renfermer’ meant ‘express’ rather than ‘contain’, although these words can often be interchanged. See for example Lagrange’s usages of the verb in ‘Ces dernières équations renferment évidemment le Principe des aires’ known today as the conservation of angular momentum, ibid., 205. Clearly, the related equations did not contain anything beyond the conservation of angular momentum (in three coordinates). See other examples on pp. 70, and 82. Poisson used the verb in a similar way, regarding, for example, the equation that express (renferme) the principle of virtual velocities, Siméon Denis Poisson, Traité de mécanique, 1811, vol 1, 233, 240, and in other contexts pp. 81, 83, 84, 154 and 401.

10 Lagrange, Mécanique analytique (n.9), 288. On 272 he wrote that the equation expresses [renfereme] ‘the principle of living forces’, on p. 239 that it ‘contains [contient] the principle of the conservation of living forces.’ On the use of T and V p. 226.

11 One may argue that at this stage the name ‘the principle of conservation of living forces’ became a kind of metonymy for an extension of the principle to realms where there is no conservation, but the basic equation (1) is still valid. This is possible, but it seems to me more plausible to regard it as pertaining to conservation of the sum of Lagrange’s T and V, and the possibility to regain the initial amount of living force.

12 Joseph-Louis Lagrange, Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits ou d’évanouissans, de limites ou de fluxions, et réduits a l’analyse algébrique des quantités finies, 1797, 266–77, quotations on 270, second 271, and 275 calculation of the living force on 275.

13 Compare Jean d’Alembert’s text in Traité de dynamique (Coignard, 1743), 169 with Joseph-Louis Lagrange in Mécanique analytique, 2^nd ed., vol. 1 (Paris, 1811), 290. Uncharacteristically, Caneva writes that d’Alembert included a discussion of living forces only in the 1758 second edition of his treatise, Caneva, ‘conservation of vis viva’ (n.3).

14 Lagrange, ibid, e.g. 291 (first with ‘conservation’ and immediately after without it), 296 (without), 299 (without), 318–9 (with).

15 Although Lazare Carnot’s Essai sur les machines general of 1783 ‘drew very little attention’ (Grattan-Guinness, Convolution in French Mathematics, 1061) it did went through a few printing and new edition (ibid., 293). It discusses the loss of living force in machine due to impact and friction, and also the connection between weight and living force, so it might have influenced Lagrange. Among others, Carnot introduced an equation for the lost living force in collisions, later known as Carnot’s theorem, which appeared in different formulation in Lagrange’s 1797 book (p. 273), and he suggested a difference equation similar to Lagrange’s. Whether influenced by Carnot or not, Lagrange did not adopt the former's use of a term for the path integral function on the force (F). See Lazare Carnot, Essai sur les machines en général, 2^nd ed., 1786, 39, 48–51. 73–4; Darrigol, ‘God, Waterwheels, and Molecules’, 308–11.

16 Lagrange introduced H instead of F in eq. (1) of the first edition as ‘an arbitrary constantequals to the value of the first member of the equation [i.e. the living forces] at a given instant’ (ibid. p. 290, the same text appear in the first ed. on p. 208).

17 Joseph Louis Lagrange, Mécanique analytique, 2^nd ed. vol. 2 (Paris, 1815), 1–35.

18 For example, in discussing in bodies subject to two central forces, ibid. 109.

19 Caneva, ‘conservation of vis viva’ (n.3), p.42, quoting Siméon Denis Poisson, Traité de mécanique, (1811), vol. 2 p. 293.

20 Poisson, ibid; the original theorem appears in vol.1 on pp. 454–5.

21 Caneva, ‘conservation of vis viva’ (n.3), p.42.

22 Pierre Simon Laplace, Traité de mécanique céleste, vol. 1 (Paris, Duprat, 1799), 52.

23 There he wrote $\sum m{v}^2-\sum m{k}^2=2\varphi (x,y,z,x^{\mathrm{\prime }},etc.)-2\varphi (a,b,c,a^{\mathrm{\prime }}[\mathrm{etc}.])$, Siméon-Denis Poisson, Traité de mécanique, 2^nd ed, 1833, vol, 2, 478.

24 Poisson, mécanique (1811) (n.19), vol. 2, 409–18 (floating body), xix, 286 (multiple references to the conservation).

25 Poisson, mécanique (1833) (n.23) 475–94 (discussion of living forces), 593–601 (floating body), 747–782 (‘Addition’). Caneva mentions that in the 2^nd edition Poisson referred to two consequences of the principle (without conservation) ‘commonly assigned to what was more usually termed the principle of the conservation of vis viva,’ Caneva, ‘conservation of vis viva’ (n.3), 43.

26 Navier mentioned that he followed Coulomb who had adopted the term ‘quantité d’action’ Claude Louis Navier’s additions to Bernard Forest de Belidor and, Architecture hydraulique ou l’art de conduire, d’élever, et de ménager les eaux pour les différents besoins de la vie, Nouv. ed. (Paris: Didot, 1819), 379, Grattan Guinness, Convolution (n. 7), 1056–8.

27 Navier in Belidor (n.26), 105–6.

28 Navier in Belidor (n.26),107–20, 290–1.

29 Navier in Belidor (n.26), 380; Darrigol, ‘God’ (n. 7), 315.

30 Yet, other comments by Navier suggest that he denied hard collisions and thus was led to extending the realm of validity of the conservation to collisions, Darrigol, ‘God’ (n. 7), 323.

31 Navier in Belidor (n.26), 376–394, especially 385. Navier referred to his previous discussion on 107-20. The report was signed by Poisson, Gérard, Fourier and de Prony and appeared as a preface to the book, vii-viii.

32 Navier in Belidor (n.26), 376.

33 Gustave Coriolis, Du calcul de l’effet des machines, ou vonsidérations sur l’emploi des moteurs et sur leur évaluation : pour servir d’introduction à l’étude spéciale des machines, 1829, p. ii (on the definition of living force); Navier in Belidor (n.26), 380. Grattan-Guinness, ‘Work’ (n. 7).

34 Gustave Coriolis, Traité de la mécanique des corps solides et du calcul de l’effet des machines, Seconde édition, 1844.

35 Jean-Victor Poncelet, Cours de mécanique industrielle fait aux artistes et ouvriers messins (1829), discussion on vii-viii. Another example of avoiding the conservation term regarding work and a principle or equation of living forces can be found in Jean Baptiste Belanger, Cours de mécanique ou résume de leçons sur la dynamique, la statique et leurs applications a l'art de l'ingénieur (1847).

36 Pontécoulant departed from Poisson in his discussion of the validity of the principle. After showing the relation between the equation and central forces he concluded: ‘Thus, the equation [of living forces], and the principle of living forces that we have deduced hold [ont lieu] in the motions of every system of bodies subject to the mutual actions and where the attractions are directed towards fixed centres, which includes more or less [à peu près] all forces of nature.’ Gustave de Pontécoulant, Théorie analytique du système du monde (Paris: Bachelier, 1829), vol 1: 97–100, quotations on 99 and 100. Here Pontécoulant got ‘more or less’ close to the announcement of complete conservation in nature. Yet the 2^nd edition of 1856 suggests that he was less close as he added that the principle does no longer hold when one of the bodies moves in a resisting medium (id., Théorie analytique du système du monde. 2^nd edition, 1856, vol. 1, 104–5). The omission of the term ‘conservation’ in the second quote just suggests an abbreviation, as Pontécoulant referred to his previous use of the term in the first quote and mentioned ‘conservation’ again on pp. 102 and 107.

37 Gustave de Pontécoulant, Traité élémentaire de physique céleste, ou précis d’astronomie théorique et pratique, seconde partie (Paris, 1840), 504.

38 Pontécoulant, Théorie analytique (n. 36), vol. 1, 131, 137–8, vol. 2, 169, 186–7, 191–8, 206–10.

39 Caneva, ‘conservation of vis viva’ (n.3), p. 21.

40 André-Marie Ampère, ‘Notice sur les nouvelles expériences électro-magnétiques faites par différents physiciens, depuis le mois de mars 1821’, Journal de physique, de chimie, d’histoire naturelle et des arts, 94 (1822), 65–66. André-Marie Ampère, ‘Extrait d'un Mémoire sur les phénomènes électrodynamiques par M. Ampère: lu le 22 décembre 1823,’ Annales de chimie et de physique, 26 (1824), 134–162, 246–258, on 256

41 André-Marie Ampère, Théorie des phénomènes électro-dynamiques, uniquement déduite de l’expérience, 1826, 124, 193, and Katzir, ‘Employment Before Formulation’ (n. 4): 16–9.

42 Up to 1824 authors usually referred to the principle of the conservation of living forces. The examples mentioned by Caneva for the name of the principle without ‘conservation’ are all from 1829 onwards, Caneva, ‘conservation of vis viva’ (n.3), 18 fn. 4.

43 Sadi Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Paris: Bachelier, 1824), pp. 21–22, translated in id, Reflexions on the Motive Power of Fire: A Critical Edition with the Surviving Scientific Manuscripts, ed. and tr. Robert Fox (Manchester: Manchester University Press, 1986), 69. Caneva claims that in the context of the denial of perpetual motion Carnot did not ‘made reference to the conservation of vis viva.’ Yet, in an endnote he adds ‘[n]ote, however, Carnot’s footnote to his argument’ and he quotes the sentences that I quote above. He adds that ‘Helmholtz . . . had not then seen Carnot’s memoir Caneva, Helmholtz (n. 3), 131, 537 (fn. 54).

44 Lazare Carnot, Essai sur les machines (n. 15), 94–6 (discussion of perpetual motion), 78 (equation of living force), 51 (1^st quote), 39 (2^nd quote). In the discussion of perpetual motion Carnot referred only indirectly to the conservation of living forces through verbal allusion to its equation and the condition that all the forces be central.

45 Fox in Carnot, Reflexions on the Motive Power (n. 43), 14, see also 122–3.

46 Sadi Carnot, Réflexions sur la puissance motrice du feu: Édition critique, ed. Robert Fox (Vrin, 1978), 247, translated in Carnot in Fox (n. 44), 191 (unlike Fox, I translated force vive as living force).

47 On the concept of hard collisions and its demise see Darrigol, ‘God’ (n. 7), especially 318–25.

48 See examples in fn. 12 and 36.

49 As mentioned above the examples mentioned by Caneva for the name of the principle/ equation without ‘conservation’ are all from 1829 onwards and from members of the engineering school plus Poisson, Caneva, ‘conservation of vis viva’ (n.3), 18 fn. 4.

50 Compare his view to that of Pontécoulant (n. 36) who suggested that all forces are central attractive forces, but in 1856 denied the universality of the principle of conservation of living force due to the resistance of the medium. Had he thought of resistance as the outcome of short-range central forces, it would have fallen in the realm of the principle. Thus, it seems that he thought about resistance as hard collisions on macroscopic bodies.

51 Caneva, Helmholtz (n. 3), e.g. 131–2, 456.

52 Caneva, ‘Conservation of vis viva’ (n.3), 55

53 Helmholtz, Über die Erhaltung der Kraft (n.3), 14.

54 Helmholtz did not refer to the authors mentioned here, except vaguely about researchers of the previous century. Yet, historian agree that he was well versed in analytical mechanics, as seen in his writings. We do not know what exactly he had read, but know that Lagrange’s fonctions analytique, for example, was in his institute’s library, Caneva, Helmholtz (n.3).

55 Katzir, ‘Employment Before Formulation’ (n. 4), Grattan Guinness, Convolution (n. 7), ch. 16, Darrigol, ‘God’ (n. 7).

56 Caneva, ‘Conservation of vis viva’ (n.3), e.g., 456–7, 359, where he suggested that Helmholtz’s omission ‘perhaps reflects the grounding of his deliberations in rational mechanics, where mathematical representations often stood in, unnamed, for absent concepts’, and passim.

57 Indeed, the term living force could also conveyed metaphysic burden as it had with Leibniz. Yet in using it the authors repeated a highly common concept. Moreover, living force was stripped off much of the metaphysics connotation by 18th century mechanics. Suggesting a new concept was another thing.

58 Notice that Lagrange deduced more physical results when he employed the symbol H for the sum in the second edition of Analytical Mechanics then in the first edition that lacked such a symbol.