On the Concept of Force: How Understanding its Historycan Improve Physics Teaching

Ricardo Lopes Coelho

Published online: 14 January 2009� Springer Science+Business Media B.V. 2009

Abstract Some physicists have pointed out that we do not know what force is. The most

common definition of force in textbooks has been criticized for more than two centuries.

Many studies have shown that the concept of force is a problem for teaching. How to

conceive force on the basis of the concepts and criticism of force in the works of Newton,

Euler, d’Alembert, Lagrange, Lazare Carnot, Saint-Venant, Reech, Kirchhoff, Mach, Hertz

and Poincare is the question of the present article. This part of the article is followed by an

overview of definitions of force in contemporary textbooks. In the next part, an answer to

the question is given: how to understand force within the framework of the laws of motion

and in applications. Finally, some educational implications are considered.

1 Introduction

In the International Congress for Philosophy in Paris, 1900, Poincare put forward the

question of if the fundamental equation of dynamics, F = ma, is verifiable experimentally.

The question itself involves, however, a problem, he said, for we do not even know what

force and mass are. In recent textbooks on mechanics, as for instance in Bergmann and

Schaefer’s, Experimental Physics (1998) or Dransfeld et al. Physics (2001), it can also be

read that we do not know what force is. If we do not know what it is, it is difficult to

explain it in the best way. As force is a fundamental concept of mechanics and mechanics

is basic in physics, it is not surprising that force is the dominant theme in the miscon-

ceptions’ literature (Carson and Rowlands 2005, p. 473).

In textbooks of the twentieth and twenty-first century, force is in general defined as the

cause of acceleration. Since acceleration is observable, its cause must be something real.

Thus, force is real. Some physicists and philosophers of science have, however, pointed out

that force does not exist in reality. Anyway, the mere fact that these two kinds of theses

coexist, shows the difficulty in ‘‘seeing’’ force in phenomena. Thus, the abstraction of force

R. L. Coelho (&)Faculty of Science, University of Lisbon, Campo Grande C4, 1749-016 Lisbon, Portugale-mail: rlc@fc.ul.pt

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Sci & Educ (2010) 19:91–113DOI 10.1007/s11191-008-9183-1

from accelerated motions must obviously be difficult for students (see Carson and Row-

lands 2005, p. 479; Rowlands et al. 2007, p. 30–31; Matthews 2008, p. 7, 10).

There was a considerable effort concerning the understanding of force. D’Alembert,

Lazare Carnot, Kirchhoff, Hertz, among others, did not only criticize the most common

definition of force but also developed new theories in order to avoid that concept.

Otherwise, there was a considerable effort as well in systematizing and applying mechanics

to new domains—Newton, Euler, Lagrange, among many others—which is connected with

that concept of force. How force could be conceived in compliance with these scientists’

contributions and without the inconveniences raised by the criticism of the concept, is the

question to deal with in the present article. To this aim, the authors who have been object

of historical studies on the concept of force (Dugas 1950; Jammer 1999; Coelho 2001) will

be considered. This part is followed by an overview of definitions of force in contemporary

textbooks (1901–2008). Finally, the connection between the concept, phenomena and

equation of force will be dealt with.

2 History

In 1687, Newton published the Mathematical Principles of Natural Philosophy. The

fundaments of his theory consist of eight definitions and three axioms. The first definition

concerns matter, the second motion and the other six concern forces. Five of the eight

propositions define quantities and the other three, concepts: innate, impressed and cen-

trifugal force (Definition III, IV, V). Centrifugal force is a particular case of impressed

force.1 Thus, according to the definitions, there are two kinds of force: innate and

impressed. Innate force is inherent in bodies2 and impressed is exterior to them.3 From the

innate kind, there is only one force, called force of inertia.4 All the others, like pressure or

impact, are impressed forces. Force of inertia justifies that a body resists change of its

motion or resting. Changes in motion require impressed forces. With these two kinds of

force the axioms are connected.

According to the first law of motion, a body perseveres in its state of resting or of

moving uniformly in a straight line, unless an impressed force constrains it to change its

state.5 The second law of motion states: ‘‘The change of motion is proportional to the

motive force impressed, and is made in the direction of the right line in which that force is

impressed’’.6 By ‘‘motion’’ is understood ‘quantity of motion’, i.e., the product of mass and

velocity of the body (Definition II). A force is double another one, Newton adds, if the

1 ‘‘Est autem vis impressa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta’’ (1726, p. 2).2 ‘‘Definitio III. Materiae vis insita est potentia resistendi, qua corpus unumquodque, quantum in se est,perseverat in statu suo vel quiescendi vel movendi uniformiter in directum ‘‘(p. 2).3 ‘‘Definitio IV. Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi velmovendi uniformiter in directum. Consistit haec vis in actione sola, neque post actionem permanet incorpore. Perseverat enim corpus in statu omni novo per solam vim inertiae’’ (p. 2).4 ‘‘Per vim insitam intelligo solam vim inertiae’’ (p. 389).5 ‘‘Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenusillud a viribus impressis cogitur statum suum mutare’’ (p. 13).6 ‘‘Lex II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectamqua vis illa imprimitur ‘‘(p. 13).

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change caused by the first is double the change caused by the second.7 Force is, therefore,

the concept for changes in the quantity of motion.

From a formal point of view, force is a deviation from the motion referred to in the first

law. As this motion is ascribed to a body on its own, force is consequently an external

action.

Euler’s Mechanics or the Science of Motion Presented Analytically, (1736), consists of

two books: the first deals with free motions and the second with constrained motions.

Euler’s approach to free motion is based on the following sequence: a body by itself stays

at rest or maintains the uniformity and rectilinearity of motion.8 Force is that which

changes these states.9 In conformity with this, Euler carries out the decomposition of force.

If a body is moving on a plane, two components of force are considered:10 tangential force,

whose effect is only the change of velocity,11 and radial force, which has no other effect

than the change of direction of the motion.12 Euler’s approach can be interpreted in the

following way. The mechanical states of a body by itself correspond to the motion of

reference. Force is a deviation from this motion. The components of force correspond to

the negations of the characteristics of the motion of reference. Let us move on to con-

strained motion.

Euler was the first to deal with the motion constrained by a surface. In this case, three

components of force are considered. The first one concerns the pressure exerted by a surface

upon a moving body. This component must exist if the motion is conditioned. The other two

components can exist or not. If they do not exist, a body covers the shortest line on the

surface and moves uniformly, says Euler. If a body does not move uniformly, then there is a

component tangential to the motion. If the body does not cover the shortest line, another

component is considered.13 Euler’s approach can be interpreted as follows. The motion of a

body constrained by a surface—it covers the shortest line uniformly—represents the motion

of reference under those circumstances. Force is a deviation from that motion. The com-

ponents of force correspond to the negation of the characteristics of this motion of reference.

A difficulty arises concerning the connection of the concept of force with the phe-

nomena. Euler deals briefly with this topic in the scholium to the definition of force (§ 102)

and in another scholium, in the second book, in the following way. It is difficult to think of

7 ‘‘Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, sive simul & semel, sivegradatim & successive impressa fuerit’’ (p. 13).8 ‘‘Corpus absolute quiescens perpetuo in quiete perseverare debet, nisi a causa externa ad motum solli-citetur’’ (Vol. I, § 56). ‘‘Corpus absolutum habens motum aequabiliter perpetuo movebitur, et eademceleritate iam antea quovis tempore fuit motum, nisi causa externa in id agat aut egerit’’ (Vol. I, § 63).‘‘Corpus absoluto motu praeditum progredietur in linea recta, seu spatium, quod describit, erit linea recta’’(Vol. I, § 65).9 ‘‘Potentia est vis corpus vel ex quiete in motum perducens vel motum eius alterans’’ (vol. I, § 99).10 ‘‘Si corpus in eodem plano moveatur in eoque etiam positae sint potentiarum sollicitantium directiones,singulae potentiae resolvi possunt in binas, quarum altera sit normalis, altera tangentialis’’ (Vol. I, § 550).11 ‘‘Vis igitur tangentialis in corpus, dum elementum Mm percurrit, alium effectum non exerit, nisi quodmotum eius vel acceleret vel retardet’’ (Vol. I, § 544).12 ‘‘In hoc vero eius effectus consistit […] ut corporis tantum directionem immutet et efficiat, ut corpus,quod per se in recta esset progressurum, in linea curva promoveatur’’ (Vol. I, § 549).13 ‘‘Prima potentia M, cuius directio in superficiem est normalis, nullum habebit effectum in immutandocorporis motu, sed tota impendetur in pressionem superficiei. […] Secunda potentia N, quia eius directio inipsa superficie est posita et normalis in directionem corporis, corporis directionem tantum immutabit ce-leritatem neque augendo neque minuendo. Haec vis igitur corpus a linea brevissima deducet facietque, utnon amplius in plano ad superficiem normali moveatur […]. Tertia potentia T, quia in directione corporis estposita, celeritatem tantum vel auget vel diminuit’’ (Vol. II, § 79).

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force without motion. Otherwise, motion can exist without force. Hence, he concludes that

all forces which we observe, have origin in motions.14 This difficulty with observing force

became the problem of the concept. Some theories of mechanics were carried out in order

to avoid that concept of force. The first of them is dealt with next.

In 1743, d’Alembert published the Traite de Dynamique, whose main contribution is a

general method of solving problems. The second part of the book, which consists of two

parts, is dedicated to the method and its applications. The first deals with the principles of

mechanics which justify the method: the principles of inertia, equilibrium and the com-

position of motion.15

The first principle states that a body maintains its rest or if moving, will move recti-

linearly and uniformly, if no external causes act on it.16 As accelerated motions are

observable, there can be no doubt concerning the existence of those causes.17 There was,

however, an objection against the use of force in mechanics.

The cause of motion was represented at that time by u in the equation udt = du, where

dt and du represent small quantities of time and velocity. D’Alembert includes this

equation in his theory: it defines accelerative force.18 However, he does not accept its

meaning.19 According to him, the thesis that force is the cause of acceleration is based on

the ‘‘vague’’ and ‘‘obscure’’ principle that the cause is proportional to the effect.20 In fact,

he continues, excepting impact, force is unknown to us.21 It is said, he exemplifies, that

weight is the cause of acceleration by falling. However, what is observed is only the

motion and not the force.22 Hence, he carried out a theory of mechanics without supposing

14 ‘‘Motum enim semel existentem perpetuo conservari debere clare ostendimus supra (§ 63); hic vero,quemadmodum ex motu potentiae oriantur, exposuimus. Quemadmodum vero potentiae sine motu velexistere vel conservari queant, concipi non potest. Quamobrem concludimus omnes potentias, quae inmundo conspiciuntur, a motu provenire’’ (Vol. II, § 29).15 ‘‘Le Principe de l’equilibre joint a ceux de la force d’inertie & du Mouvement compose, nous conduit a lasolution de tous les Problemes […]’’ (1758, p. xv).16 ‘‘Un Corps en repos y persistera, a moins qu’une cause etrangere ne l’en tire’’ (p. 3–4). ‘‘Un Corps misune fois en mouvement par une cause quelconque, doit y persister toujours uniformement & en ligne droite,tant qu’une nouvelle cause, differente de celle qui l’a mis en mouvement, n’agira pas sur lui’’ (p. 4).17 ‘‘On appelle en general puissance ou cause motrice, tout ce qui oblige un Corps a se mouvoir’’ (p. 4).‘‘Cette variation continuelle ne peut provenir (art. 6.) que de quelque cause etrangere qui agit sans cesse,pour accelerer ou retarder le Mouvement’’ (p. 17).18 ‘‘nous nous contenterons […] d’entendre seulement par le mot de force acceleratrice, la quantite alaquelle l’accroissement de la vitesse est proportionnel’’ (p. 25).19 ‘‘La plupart des Geometres presentent sous un autre point de vue l’equation udt = du entre les tems &les vitesses. Ce qui n’est, selons nous, qu’une hypothese, est erige par eux en principe. Comme l’ac-croissement de la vitesse est l’effet de la cause acceleratrice, & qu’un effet, selon eux, doit etre toujoursproportionnel a sa cause, ces Geometres ne regardent pas seulement la quantiteu comme la simpleexpression du rapport de du a dt; c’est de plus, selon eux, l’expression de la force acceleratrice, a laquelle ilspretendent que du doit etre proportionnel, dt etant constant’’ (p. 24–25).20 ‘‘Pourquoi donc aurions-nous recours a ce principe dont tout le monde fait usage aujourd’hui, que la forceacceleratrice ou retardatrice est proportionnelle a l’element de la vitesse? principe appuye sur cet uniqueaxiome vague & obscur, que l’effet est proportionnel a sa cause. […] nous nous contenterons d’observer,que […] il est inutile a la Mechanique, & que par consequent il doit en etre banni’’ (p. xii).21 ‘‘Le Mouvement uniforme d’un Corps ne peut etre altere que par quelque cause etrangere. Or de toutesles causes, soit occasionnelles, soit immediates, qui influent dans le Mouvement des corps, il n’y a tout auplus que l’impulsion seule dont nous soyons en etat de determiner l’effet par la seule connoissance de lacause, comme on le verra dans la seconde Partie de cet Ouvrage. Toutes les autres causes nous sontentierement inconnues’’ (p. 22).22 ‘‘toutes les autres causes ne se font connoıtre que par l’effet, & nous en ignorons entierement la nature:telle est la cause qui fait tomber les Corps pesans vers le centre de la Terre’’ (p. xi).

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a knowledge of the nature of force. He did not presume, for instance, that forces act

together as each of them would act by itself.23 He further argues that the knowledge of

motion is enough for science and opted to work out a theory starting from matter and

motion.24

In sum, d’Alembert admitted force from an ontological point of view but not as an

object of knowledge due to the lack of observability.

Euler in 1750, published an article with the title ‘‘Discovery of a New Principle

of Mechanics’’. The new principle is only an equation, whose form is ‘force =

mass � acceleration’.25 Force, which he also called ‘‘potentia’’ (1736), is decomposed into

three components, symbolized by Px, Py, and Pz. The components have the form Px = 2 md2x/dt2.26 With this kind of decomposition, the information concerning the path covered by

a body and how it is covered lies in the coordinates and in the acceleration along the

coordinates.

Lagrange’s Analytical Mechanics, 1788, made a new contribution to the concept of

force and to the decomposition of motion. This book consists of two parts: statics and

dynamics. Statics is defined as the science of the equilibrium of forces27 and dynamics as

the science of forces and of the motions which are caused by them.28 The concept of force

is presented at the beginning of the first part, as the cause or tendency to cause motion in a

body.29 This meaning will change in the course of the development of the theory. This

development is connected with the proposition which unifies the theory, and will be

considered briefly: the addition of the moments of force equals zero.

In statics, two forces are in equilibrium if their values, P and Q, and distances to the

fulcrum, dp and dq, are related as follows: P dp = -Q dq. The general equation for the

23 ‘‘Quelques Lecteurs pourront etre surpris de ce que je tire la demonstration d’une proposition si simple enapparence, d’un cas general beaucoup plus compose; mais on ne peut, ce me semble, demontrer autrement laproposition dont il s’agit ici, qu’en regardant comme un axiome incontestable, que l’effet de deux causesconjointes est egal a la somme de leurs effets pris separement, ou que deux causes agissent conjointementcomme elles agiroient separement; principe qui ne me paroıt pas assez evident, ni assez simple, qui tientd’ailleurs de trop pres a la question des forces vives, & au principe des forces acceleratrices dont nous avonsparle ci-dessus art. 22. C’est la raison qui m’a oblige a eviter d’en faire usage’’ (p. 38–39).24 ‘‘A l’egard des demonstrations de ces Principes en eux-memes, le plan que j’ai suivi pour leur donnertoute la clarte & la simplicite dont elles m’ont paru susceptibles, a ete de les deduire toujours de laconsideration seule du Mouvement, envisage de la maniere la plus simple & la plus claire. Tout ce que nousvoyons bien distinctement dans le Mouvement d’un Corps, c’est qu’il parcourt un certain espace, & qu’ilemploye un certain tems a le parcourir. C’est donc de cette seule idee qu’on doit tirer tous les Principes de laMechanique, quand on veut les demontrer d’une maniere nette & precise’’ (p. xvi). ‘‘De toutes cesreflexions, il s’ensuit que les loix de la Statique & de la Mechanique, exposees dans ce Livre, sont celles quiresultent de l’existence de la matiere & du mouvement’’ (p. xxviii).25 ‘F = ma’ is usually called ‘Newton’s second law’. According to Euler, who read the Principia, Newton’ssecond axiom is not expressed by that equation. As a matter of fact, that equation does not appear anywherein Newton’s Principia. Moreover, historians of science do not agree with each other concerning an equationfor Newton’s second law. (See Cohen 1970, p. 144; Dellian 1985, p. 401; Maltese 1992, p. 26).26 Euler writes the equation firstly to coordinate x and afterwards similarly for y and z. ‘‘Apres l’element dutems dt, soit x ? dx la distance du corps au plan et prenant cet element dt pour constant, il sera 2Mddx ¼�P dt2; selon que la force P tend ou a eloigner ou a approcher le corps du plan. Et c’est cette formule seule,qui renferme tous les principes de la Mecanique’’ (Opera Omnia, Ser. II, Vol. 5, p. 89).27 ‘‘La Statique est la science de l’equilibre des forces’’ (1888–1889, Vol. I, p. 1).28 ‘‘La Dynamique est la science des forces acceleratrices ou retardatrices et des mouvements variesqu’elles doivent produire’’ (Vol. I, p. 237).29 ‘‘On entend, en general, par force ou puissance la cause, quelle qu’elle soit, qui imprime ou tend aimprimir du mouvement au corps auquel on la suppose appliquee’’ (Vol. I, p. 1).

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equilibrium of two forces can, therefore, be written in the form P dp ? Q dq = 0. P dp is

called ‘moment of force’ and Q dq as well.30 Thus, that equation can be read: the sum of

the moments of forces is equal to zero. Equations with three or more forces in statics as

well as the equations of motion can be read in the same way. The general equation of

dynamics in generalised coordinates, for instance, is written in the form

NdnþWdwþ . . . ¼ 0

where ‘‘Ndn’’ is a ‘moment of force’. The elements of ‘moment of force’ have been

generalised.31 This generalisation might obscure the meaning of force.

Let us consider a simple example to clarify the interpretation: the circular movement of

a point around an axle. Supposing that the motion takes place on the plane XY and taking

the angle h between the X-axis and the radius as the generalised coordinate, a short piece of

the path ds is given by r dh. Once the geometrical element of a motion is determined,

Lagrange’s equations indicate how the path is covered. The two possible forms are

mr2€h ¼ 0

or

mr2€h ¼ Qh :

In the first case, the motion is uniform; in the second, there is a deviation from

uniformity.

From a formal point of view, the circular uniform motion functions here as the motion

of reference. There is place for force only if that motion changes.

Lazare Carnot developed a new theory of mechanics in order to avoid the concept of

force as the cause of acceleration. According to his Principes fondamentaux du mouvementet du repos, (1803), there are two ways of carrying out mechanics: either as a theory of

force or as a theory of motion.32 The first one was followed by almost all the authors, said

Carnot. He also acknowledged its advantages. It has, however, one shortcoming, being

based on the ‘‘metaphysical’’ concept of force. This gave him the reason for opting for the

second method.33 The problem pointed out by Carnot concerns the observation of force

30 ‘‘Nous nommerons chaque terme de cette formule, tel que Pdp, le moment de la force P […] la formulegenerale de la Statique consistera dans l’egalite a zero de la somme des moments de toutes les forces’’ (Vol.I, p. 29–30).31 Some examples of generalization. 1.’’Nommons E la force de l’elasticite et e l’angle exterieur qu’elletend a diminuer; le moment de cette force sera exprime par Ede (Sect. II, art. 9), de sorte que la somme desmoments de toutes les forces du systeme sera […] ? E de.’’ (Vol. I, p. 143). 2. ‘‘Appliquons les memesprincipes a la determination de l’equilibre d’une surface dont tous les elements dm soient extensibles etcontractibles […]’’ (See Vol. I, p. 158–159). 3. ‘‘A l’egard de la quantite k dont nous venons de determinerla valeur, il est bon de remarquer que le terme Sk dL de l’equation generale de l’article 10 represente lasomme des moments d’autant de forces k qui tendent a diminuer la valeur de la fonction L […]’’ (See Vol. I,p. 214–215).32 ‘‘Il y a deux manieres d’envisager la mecanique dans ses principes. La premiere est de la considerercomme la theorie des forces, c’est-a-dire des causes qui impriment les mouvemens. La seconde est de laconsiderer comme la theorie des mouvemens eux-memes’’ (p. xi).33 ‘‘La premiere methode offre donc beaucoup plus de facilite; aussi est-elle, comme je l’ai observe ci-dessus, presque generalement suivie’’ (p. xv–xvi). ‘‘La premiere est presque generalement suivie, comme laplus simple; mais elle a le desavantage d’etre fondee sur une notion metaphysique et obscure qui est celledes forces’’ (p. xi–xii). ‘‘j’ai adopte ici la seconde comme je l’avois deja fait dans la premiere edition; parceque j’ai voulu eviter la notion metaphysique des forces’’ (p. xvi).

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and was presented in considering machines, according to the author, as the most important

object of mechanics.

Some machines from that time worked thanks to men or animals. If a human being

brings a machine into motion, he is the cause of that motion. The cause of motion was

force, according to science. Is this force, Carnot questioned, the structure of the skeleton of

the human being or of an animal or their wills? Does a double force mean, he continued to

ask, that the will in the first case is double that in the other?34 This questioning shows the

difficulty.35

As a solution to this problem, Carnot proposed to identify force with the quantity of

motion which a force caused in a body.36 In doing this, we do not know more about the

force which causes motion but it does not disturb the theory.37 Within this, force is a

certain quantity of motion or, in other words, the motion caused by real force, called ‘‘first

cause’’.

In the introduction to the book, Carnot defends the thesis that what we know comes

from experiments.38 From them, Carnot drew 7 statements called hypotheses, which

constitute the starting point of his theory.39 The first of them corresponds to the law of

inertia.40 Once admitted that a body by itself maintains its resting or moving rectilinearly

and uniformly, it follows that whatever motion requires an external cause. The ‘‘first

cause’’ satisfies this requirement. It does not satisfy, however, the epistemological

requirement of observation.

Barre de Saint-Venant also carried out a reorganization of mechanics in Principles ofMechanics, (1851). This book is divided into three thematic domains: kinematics,

dynamics and statics. In kinematics, the motion is considered ‘‘merely geometric’’. It is

34 ‘‘Ces causes sont-elles la volonte ou la constitution physique de l’homme ou de l’animal qui par sonaction fait naıtre le mouvement? Mais qu’est-ce qu’une volonte double ou triple d’une autre volonte, ou uneconstitution physique capable d’un effet double ou triple d’une autre?’’ (p. xii).35 ‘‘quelle idee nette peut presenter a l’esprit en pareille matiere le nom de cause? il y a tant d’especes decauses! Et que peut-on entendre dans le langage precis des mathematiques par une force, c’est-a-dire, parune cause double ou triple d’une autre?’’ (p. xii).36 ‘‘Si l’on prend le parti de ne point distinguer la cause de l’effet, c’est-a-dire, si l’on entend par le motforce la quantite de mouvement meme qu’elle fait naıtre dans le mobile auquel elle est appliquee, on devientintelligible’’. (p. xii-xiii). ‘‘Je repeterai d’abord, qu’il ne s’agit point ici des causes premieres qui font naıtrele mouvement dans les corps, mais seulement du mouvement deja produit et inherent a chacun d’eux. C’estcette quantite de mouvement deja produite dans un corps, qu’on nomme sa force ou sa puissance’’ (p. 47).‘‘ainsi que nous l’avons deja observe, on ne considere, en mecanique, aucune force qui ne reside effec-tivement dans les corps, c’est-a-dire, qui ne soit reellement une quantite de mouvement deja produite’’ (p.108).37 ‘‘La mecanique ne remonte pas jusqu’aux causes premieres qui produisent le mouvement; elle n’examinepas comment la volonte de l’homme ou de l’animal fait sortir ses membres du repos, ou les y ramenespontanement: elle ne voit que le fait qui en resulte, ne considere que le mouvement deja produit, et sonobjet est uniquement de rechercher comment ce mouvement une fois imprime, se conserve, se propage ou semodifie’’ (p. 33).38 ‘‘Les anciens etablirent en axiome que toutes nos idees viennent des sens: et cette grande verite n’est plusaujourd’hui un sujet de contestation. Il suit de-la, que toute science quelconque tire ses elemens de l’ex-perience, puisque les premieres idees qu’elle puisse combiner sont le resultat de nos sensations, qui ne sontautre chose que les donnees de l’experience. ‘‘D’ou l’homme tire-t-il, dit Locke, tous ces materiaux qui sontcomme le fond de tous ses raisonnemens et de toutes ses connoissances? Je reponds en un mot, del’experience’’ (p. 2).39 ‘‘On pourra remarquer que ces hypotheses rentrent en partie les unes dans les autres: mon objet n’a pasete de les reduire au plus petit nombre possible; il me suffit qu’elles ne soient point contradictoires etqu’elles soient clairement entendues’’ (p. 47).40 ‘‘Cette hypothese est le principe connu sous le nom de loi d’inertie’’ (p. 53).

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considered from a physical point of view in dynamics, according the author. Statics is

presented as a special case of dynamics. His dynamics presents a new sequence of the

concepts of force, mass, and acceleration.

Barre’s theory starts from a unique proposition, which states that the acceleration of

bodies depends on the points which constitute them. The number of points is considered

proportional to the mass of a body.41 The mass of a body is, however, not determinable by

those points. For the measurement of mass, the reciprocal alteration of the velocities of

bodies by impact is proposed.42 As the use of this measurement process is very difficult,

Saint-Venant recommended measuring mass by weighing.43 Force is defined as the product

of mass by acceleration.44

At the beginning of the Principles and again in dynamics, the problems concerning

force as cause of acceleration are referred to. The author aims to overcome those diffi-

culties in making force a mere mathematical concept. However, other difficulties arose: not

only concerning the process of measurement of mass but also the interpretation of phe-

nomena. The terms used in the dealing with phenomena remind us of the traditional

concept of force: ‘acting forces’, ‘force acts on bodies’, and analogous expressions.45

Saint-Venant was aware of this: he indicated at the beginning of dynamics, how the

traditional interpretation of phenomena is to be understood in compliance with the planned

conceptual framework.46

In sum, Barre’s aim was to make of force a mathematical concept due to the problems

caused by the concept in mechanics. For this reason, he starts with acceleration and defines

mass through the impact of two bodies. This measurement process as well the interpre-

tation of phenomena caused difficulties.

41 ‘‘On donne le nom de Masses a des nombres proportionnels a ceux des points elementaires qu’il fautsupposer dans les corps, comparativement les uns aux autres, pour expliquer leurs divers mouvemens parcette loi [la loi generale], conformement a son enonce’’ (§ 81).42 ‘‘La masse d’un corps est le rapport de deux nombres exprimant combien de fois ce corps et un autrecorps choisi arbitrairement et constamment le meme, contiennent de parties qui, etant separees et heurteesdeux a deux l’une contre l’autre se communiquent, par le choc, des vitesses opposees egales’’ (§ 81).43 ‘‘Mais on peut, en general, se dispenser de ces mesurages de vitesse et d’acceleration, qui sont delicateset difficiles, et estimer promptement les masses […] par le pesage’’ (§ 88). ‘‘Les poids des corps sont,comme l’on voit, en un meme lieu, proportionells aux masses’’ (§ 89).44 ‘‘La force ou l’action attractive ou repulsive d’un corps sur un autre est une ligne ayant pour grandeur leproduit de la masse de celui ci par l’acceleration moyenne de ses points vers ceux du premier et pourdirection celle de cette acceleration’’ (§ 81).45 See §§ 83, 85, 86, 93, 97, 98, 100, 103, 109, 116, 119, 120, 129, 138, 145, 157, 159, 161, 163, 164, 166,167, 168, 171, 172, 173, 174, 175, 178, 179, 184, 185. Some examples:—‘‘l’acceleration g qu’ils [les poids]donnent aux masses sur lesquelles ils agissent’’ (§ 93);—‘‘Si les forces agissant sur le systeme se fontequilibre […]’’ (§ 119). ‘‘Force’’ or ‘‘puissance’’ appear also with the verbs:—‘‘solliciter’’, §§ 97, 120, 159,169, 172;—‘‘appliquer’’, §§ 168, 169, 185;—‘‘exercer’’, §§ 144, 179, 185.46 ‘‘La denomination de force ou d’action vient du sentiment de l’effort que nous exercons lorsque nousvoulons imprimer une acceleration a un corps et de ce que, dans le langage commun, l’on attribuemetaphoriquement une activite analogue a celle de l’homme, aux autres etres, meme inanimes, dans ladirection desquels l’on voit des corps prendre un mouvement. Pour nous conformer a cette maniere de parlerqui a passe dans la science, nous dirons quelque fois qu’un corps A est sollicite par une force de grandeur F,emanant d’un autre corps B, et qui, en agissant sur A dans une certaine direction, produit une acceleration j oudonne a A une vitesse jt dans le temps t. Mais, par la, nous voudrons dire simplement que les points du corpsA ont, vers ceux du corps B, des composantes d’acceleration dont la moyenne a une certaine direction et unegrandeur qui, multipliee par la masse m de A, donne un produit mj egal a F. Nous dirons que nous appliquonsune force F a un corps A dans une certaine direction: cela signifiera que nous placons un ou plusieurs autrescorps animes ou inanimes dans des situations ou dans un etat physique tels que les accelerations des points deA vers leurs points aient une moyenne qui, multipliee par la masse de A, donne F’’ (§ 82).

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In the following year, Reech published the Cours de Mecanique, whose aim was,

however, very different. This book consists of two parts: first, ‘on the alleged science of

mechanics’ and second, ‘on the true science of mechanics’. The reason for such a dis-

tinction lies in the concept of force. Criticised by him is the ‘relative concept of force’. By

this is understood the measurement of force through its ‘evident geometric effect’, i.e., by

the deviation from a certain motion. Instead, Reech proposes the ‘real, absolute concept of

force’. This reality is connected with the sensation of our muscles.

Our sensation awakened in us the idea of a certain quantity, called pressure or traction,

which is the cause of the alteration of the motions of bodies which have been touched. This

is the true idea of force that we should have, says Reech in the introduction to the Course.47

In the corpus of the book, force is defined as pressures or tractions that we can make

through our organs on the bodies surrounding us.48 The process of measurement proposed

for forces reflects the definition. Force is to be measured through a convenient thread,

whose changes in length indicate the magnitude of force.49 In this process, some diffi-

culties lie inherently.

Reech pointed out that a thread has some mass and this influenced the measurement of

force. He proposed then, making a conceptual distinction between matter and connections

of material points and to ascribe mass only to matter. In doing so, the thread is considered

massless.50 Another difficulty concerns the limitations in using a thread to measure force,

as, for instance, in the case of celestial motions. For such cases, he proposes measuring

force thanks to the deviation from a certain, conventional motion. To play this role, he

chooses the rectilinear and uniform motion, not because it is the natural one, he points out,

but only because it is the simplest motion and the most commonly used one. The law of

inertia is, according to him, a mere convention.51

In 1876, Kirchhoff published a textbook on Mechanics, which became very successful;

the second edition occurred in the same year. The preface to the book announces a

restructuring of mechanics, whose leitmotiv lies in the concept of force. Physicists

47 ‘‘La seule et veritable idee que nous devions nous faire de la force, c’est celle que nous acquerons quand,a l’aide de nos organes, nous cherchons a modifier l’etat de repos ou de mouvement des corps qui nousenvironnent. Nous eprouvons alors des sensations qui eveillent en nous plusieurs idees fondamentales:d’abord celle de l’existence des corps, puis celle de la forme des corps et des proprietes de l’espace, puiscelle du mouvement et du temps, puis encore celle d’une certaine quantite que nous nommons une pressionou une traction. Cette quantite est une cause de mouvement ou plutot une cause de changement demouvement pour les parties des corps que nous rencontrons a l’aide de nos organes’’ (p. 37).48 ‘‘Par le mot force, on ne doit entendre que les pressions ou tractions que nous pouvons faire a l’aide denous organes, sur les corps qui nos environnent’’ (p. 57).49 ‘‘La direction de la force sera celle du fil dans lequel elle residera, et l’intensite de la force dependra del’allongement ainsi que de la nature du fil’’ (p. 46).50 ‘‘Par une abstraction de notre entendement, nous pouvons nous representer un fil tendu, comme etantcompletement depourvu de sa qualite matiere ou masse, et alors un pareil fil sera parfaitement indifferent ase mouvoir d’une maniere plutot que d’une autre, c’est-a-dire qu’un pareil fil suivra spontanement les corpsou obstacles, auxquels il se trouvera attache, en faisant de la force aux points d’attache sur ces obstacles, eten n’exigeant aucune force pour participer a leur mouvement’’ (p. 59).51 ‘‘Mais alors, il y aura une convention a faire. Il s’agira de savoir quelle sorte de mouvement, rectiligne oucurviligne, uniforme ou varie, nous devrons admettre, comme etant celui d’un point materiel entierementlibre en apparence, et parce que nous aurons une entiere latitude a cet egard, ainsi que nous l’avons deja faitpressentir dans la derniere section de la premiere partie, avec le seul avantage ou inconvenient d’en voirresulter de plus ou moins grandes simplifications dans les relations mecaniques des systemes, nous seronsconduits naturellement a faire servir a un tel usage l’etat de mouvement rectiligne uniforme, et a rencontrercette fameuse loi d’inertie de la matiere, qui ne sera plus un principe ni un fait d’experience, mais une pureconvention, la plus simple de toutes celles parmi lesquelles nous nous trouverons obliges de choisir’’ (p. 49).

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disagreed with each other, according to Kirchhoff, about if some important statements,

such as if the law of inertia and the parallelogram of forces were axioms, theorems or

experimental results. According to the author, these problems lie in the concept of force: in

the lack of clarity of ‘cause’ motion and ‘tendency to cause’ motion.52 To avoid these

problems, Kirchhoff restricts the function of the science to the description of motion.53

At the beginning of the book, mechanics is defined as the science of motion. As in order

to conceive motion, the notions of space, time and matter are necessary and sufficient,

according to the author, these become the primitive notions of the science of motion. Force

and mass are to be constructed within the theory.54 This plan was, however, not carried out

successfully. Kirchhoff himself detected the following difficulty. If a system of forces acts

on a body, it is impossible to determine that system only through that motion. Thanks to the

observation of that motion, we achieve the resultant but not the components of force. There

are, therefore, forces which cannot be subsumed by the theory.55

Another difficulty concerns the interpretation of phenomena. The most common

expressions connected with ‘force’ are: ‘forces act’, ‘acting forces’, ‘forces are exerted’,

‘exerting forces’.56 If forces act, they must be something which have an influence. Thus, this

terminology leads us to think of force as a cause, which Kirchhoff had planned to avoid.

In sum, Kirchhoff changed the status of force, from a real thing to a mere theoretical

concept. There was, however, a difficulty in carrying out that transformation, namely in

obtaining every force through motion and in the interpretation of phenomena.

Mach’s Mechanics was published in 1883, with successive editions and reprints. His

solution for force was taken up from a short paper written in 1868. Here, he criticized the

vicious circle in defining mass at that time: weight was defined by mass and mass by

weight.57 He proposed, then, a solution which presents the sequence: acceleration, mass,

force.

The starting proposition of Mach’s proposal, presented anew in Mechanics (1933), says

that bodies in interaction cause reciprocal acceleration.58 This is considered a matter of

52 ‘‘Man pflegt die Mechanik als die Wissenschaft von den Kra ften zu definiren, und die Krafte als dieUrsachen, welche Bewegungen hervorbringen oder hervorzubringen streben. Gewiss ist diese Definition[…] Aber ihr haftet die Unklarheit an, von der die Begriffe der Ursache und des Strebens sich nicht befreienlassen. Diese Unklarheit hat sich z. B. gezeigt in der Verschiedenheit der Ansichten daruber, ob der Satz vonder Tragheit und der Satz vom Parallelogramm der Krafte anzusehen sind als Resultate der Erfahrung, alsAxiome oder als Satze, die logisch bewiesen werden konnen und bewiesen werden mussen’’ (1897, p. V).53 ‘‘Aus diesem Grunde stelle ich es als die Aufgabe der Mechanik hin, die in der Natur vor sich gehendenBewegungen zu beschreiben, und zwar vollstandig und auf die einfachste Weise zu beschreiben’’ (p. V).54 ‘‘Zur Auffassung einer Bewegung sind die Vorstellungen von Raum, Zeit und Materie nothig, aber auchhinreichend. Mit diesen Mitteln muss die Mechanik suchen, ihr Ziel zu erreichen, und mit ihnen muss sie dieHulfsbegriffe construiren, die sie dabei nothig hat, z. B. die Begriffe der Kraft und der Masse’’ (p. 1).55 ‘‘Es ist einleuchtend, dass, wenn man eine bestimmte Bewegung eines Punktes als bedingt durch mehrereKrafte ansieht, diese nicht einzeln bestimmt sind; nur die Resultante ist bestimmt […] Aus der Bewegungallein kann die Mechanik nach unserer Auffassung die Definitionen der Begriffe schopfen, mit denen sie eszu thun hat. Es folgt daraus, dass nach Einfuhrung von Kraftesystemen an Stelle einfacher Krafte dieMechanik ausser Stande ist, eine vollstandige Definition des Begriffs der Kraft zu geben’’ (p. 11).56 See for instance, pp. 8, 13, 22, 23, 25, 30, 31, 33, 34, 35, 36, 38, 39, 45, 51, 56, 60, 62, 68, 86, 88, 89, 109,110, 115, 126, 127, 128, 132, 144, 146, 150, 160, 164, 165, 170, 171, 233, 235, 236, 244, 247, 249, 290, 308,348, 349, 352, 358, 369, 377, 385, 393, 404, 416, 418, 436, 455, 458.57 ‘‘Man definirt gewohnlich m = p/g und wiederum p = mg’’ (1868, p. 356).58 ‘‘Die Definition [der Masse] berucksichtigt lediglich die Tatsache, daß in Wechselbeziehung stehendeKorper, ob sogenannte Fernwirkungen, starre oder elastische Verbindungen in Betracht kommen, anein-ander Geschwindigkeitsanderungen (Beschleunigungen) bestimmen. Mehr als dies braucht man nicht zuwissen, um mit voller Sicherheit und ohne Furcht, auf Sand zu bauen, definieren zu konnen’’ (1933, p. 261).

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fact. Taking one body as a unit, Mach continues, the mass of the other is measured through

the proportion of the accelerations due to the interaction of both bodies.59 Force is then

defined as the product of mass and acceleration.60

Mach’s aim was also to make of force a mere theoretical concept. How to think of force

remained, however, a problem. Force is any circumstance of which the consequence is

motion, says Mach.61 This leads to the idea that force is the cause of motion. In a second

step, Mach made recourse to our sensation to understand force: the circumstances deter-

minative of motion which are best known to us, are our volitional acts. Thus, continues

Mach, our habit of representing circumstances determinative of motion as something akin

to volitional acts arises.62 He knew that this was not scientific but he did not know of a

better way: the attempts to set aside this conception as subjective and unscientific, he said,

fail invariably.63

Even though Mach gave a new definition of force and proposed to understand it as a

mere theoretical concept, the approach to phenomena was marked by the traditional

interpretation.

Hertz’s (1894) posthumous work, The Principles of Mechanics, was published. In the

introduction to the book, his philosophy of science is presented. On it is based his

mechanical theory. With this modus procedendi the author aims to overcome some

problems of mechanics. Among the main difficulties of this science is the concept of

force.

If we swing a stone tied to a piece of string in a circle, exemplifies Hertz, we are

conscious of exerting a force upon the stone. This agrees with the definition of force: force

is independent of motion and the cause of it. Newton’s third law, he continues, requires,

however, an opposing force to the force exerted by the hand upon the stone. Here the

59 ‘‘Ist uns aber einmal durch mechanische Erfahrung die Existenz eines besondern beschleunigungbest-immenden Merkmals der Korper nahegelegt, so steht nichts im Wege, willkurlich festzusetzen: Korper vongleicher Masse nennen wir solche, welche aufeinander wirkend sich gleiche entgegengesetzte Beschleu-nigungen erteilen. Hiermit haben wir nur ein tatsachliches Verhaltnis benannt. Analog werden wir in demallgemeinern Fall verfahren. Die Korper A und B ([…]) erhalten bei ihrer Gegenwirkung beziehungsweisedie Beschleunigungen -u und ?u, wobei wir den Sinn derselben durch das Zeichen ersichtlich machen.Dann sagen wir, B hat die -u/u fache Masse von A. Nehmen wir den Vergleichskorper A als Einheit an, soschreiben wir jenem Korper die Masse m zu, welcher A das mfache der Beschleunigung erteilt, die er inGegenwirkung von A erhalt. Das Massenverhaltnis ist das negative umgekehrte Verhaltnis der Gegenb-eschleunigungen’’ (1933, p. 211–212).60 ‘‘Bewegende Kraft ist das Produkt aus dem Massenwert eines Korpers in die an demselben bestimmteBeschleunigung’’ (1933, p. 242).61 ‘‘Die Kraft ist also ein bewegungbestimmender Umstand dessen Merkmale sich in folgender Art angebenlassen. Die Richtung der Kraft ist die Richtung der von der gegebenen Kraft allein bestimmten Bewegung.Der Angriffspunkt ist derjenige Punkt, dessen Bewegung auch unabhangig von seinen Verbindungen bes-timmt ist. Die Große der Kraft ist das Gewicht, welches, nach der bestimmten Richtung (an einer Schnur)wirkend, an dem gegebenen Punkt angreifend, dieselbe Bewegung bestimmt oder dasselbe Gleichgewichterhalt’’ (1933, p. 75).62 ‘‘Diejenigen bewegungbestimmenden Umstande, die uns am besten bekannt sind, sind unsere eigenenWillensakte, die Innervationen. Bei den Bewegungen, welche wir selbst bestimmen, sowie bei jenen, zuwelchen wir durch außere Umstande gezwungen sind, empfinden wir stets einen Druck. Dadurch stellt sichdie Gewohnheit her, jeden bewegungbestimmenden Umstand als etwas einem Willensakt Verwandtes undals einen Druck vorzustellen’’ (1933, p. 74).63 ‘‘Die Versuche, diese Vorstellung als subjektiv, animistisch, unwissenschaftlich zu beseitigen, mi-ßglucken uns immer. Es kann auch nicht nutzlich sein, wenn man seinen eigenen naturlichen GedankenGewalt antut und sich zu freiwilliger Armut derselben verdammt’’ (1933, p. 74).

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problem begins: ‘‘In our laws of motion, force was a cause of motion, and was present

before motion. Can we, without confusing our ideas, suddenly begin to speak of forces

which arise through motion, which are a consequence of motion?’’ (1899, p. 6). Since force

is defined as the cause of motion and there is a force that is a consequence of motion, the

theory is not logically permissible, according to Hertz. His own solution for force appears

in the following context.

According to Hertz’s philosophy of science, a physical theory is an image which we

form of things.64 His image starts from one axiom, which is formulated for free systems. A

non-free system is understood as a part of a free system,65 which is not completely

known.66 Hertz imagined, then, connections between the sub-systems and expressed them

mathematically. To verify if the consequences of the image are conform with the

respective phenomena, measurements must be made. The methods of determining force

indicated by Hertz were common in the mechanics of that time (§§ 542–544). However,

between these measurement processes and the constructed concept of force there is only a

mere correspondence.67 The meaning of force in the theory is connected with its axiom.

Hertz’s theory is based on one unique axiom called fundamental law. This proposition,

the only one drawn from experiments, according to the author, has the form of the law of

inertia: ‘‘Every free system persists in its state of rest or of uniform motion along the

straightest path’’.68 In Hertz’s mechanics, there is a force if the motion is not uniform or the

curvature of the path is not a minimum.69 Thus, force is a deviation from the motion

referred to in the fundamental law.

In sum, Hertz’s solution for force consists of a separation between force in thought,

which belongs to the image, and force in practice, which is a measurement process. From a

formal point of view, force is a deviation from the motion of the fundamental law.

Poincare (1897) wrote an article about Hertz’s mechanics, in which he asserts categor-

ically ‘‘to say that force is the cause of acceleration is to do metaphysics’’.70 He defends

instead that a concept of force should be worked out from its measurement process. Hence,

he began to consider the definition of equal forces: two forces are said to be equal if they

attain equilibrium or produce the same acceleration on the same mass. Poincare comments:

we cannot connect and disconnect forces to or from bodies as horses to coaches or engines to

carriages. It was said as well that two forces are equal if they balance with the same weight.

Poincare pointed out that the weight depends on the place. Furthermore, Newton’s third law

64 ‘‘Wir machen uns innere Scheinbilder oder Symbole der außeren Gegenstande, und zwar machen wir sievon solcher Art, daß die denknotwendigen Folgen der Bilder stets wieder die Bilder seien von den natur-notwendigen Folgen der abgebildeten Gegenstande’’ (p. 1).65 ‘‘Nach unserer Auffassung ist jedes unfreie System Teil eines großeren freien Systems’’ (§ 429).66 ‘‘Indem wir einen Teil eines freien Systems als unfreies System behandeln, setzen wir voraus, daß dasubrige System uns mehr oder weniger unbekannt ist’’ (§ 430).67 ‘‘Durch Anwendung einer jeden dieser drei Methoden konnen auch die Krafte aus Rechnungsgroßen zuGegenstanden der unmittelbaren Erfahrung gemacht werden, d.h. zu Zeichen fur bestimmte Verbindungensinnlicher Empfindungen und Wahrnehmungen’’ (§ 541).68 ‘‘Jedes freie System beharrt in seinem Zustande der Ruhe oder der gleichformigen Bewegung in einergeradesten Bahn’’ (§ 309).69 See § 368 (the differential equations of the motion of a free system) and § 482 (the equations of motionof a system influenced by forces).70 ‘‘Quand on dit que la force est la cause d’un mouvement, on fait de la metaphysique, et cette definition, sion devait s’en contenter, serait absolument sterile. Pour qu’une definition puisse servir a quelque chose, ilfaut qu’elle nous apprenne a mesurer la force; cela suffit d’ailleurs, il n’est nullement necessaire qu’elle nousapprenne ce que c’est que la force en soi, ni si elle est la cause ou l’effet du mouvement’’ (1897, p. 734).

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is employed, in such cases, as a definition and not an experimental law. Due to all these

problems, Poincare moved on to the other possibility of defining force, thanks to mass.

To this kind of sequence of definitions—acceleration, mass, force—Poincare raised

some objections as well. One of them concerns the supposition that the acceleration of

body A is caused by B, when it is caused not only by B, but also by that of C, D, E, etc.71

To determine the mass of A based on the acceleration produced by B, it would be nec-

essary to separate A’s acceleration into its elements.72 This decomposition of the

acceleration would, however, only be possible, according to Poincare, if the hypothesis of

central forces was admitted.73 As the hypothesis does not offer guarantees, Poincare goes

on to another possibility, the determination of mass through the center of mass.74

The center of mass of a system, on which no exterior action is exercised, is charac-

terized by a uniform and rectilinear motion. Thus, the values of masses of bodies could be

combined in such a way that the center of mass might show such a motion. Since there is

not a system without exterior action, the law of the movement of the center of mass would

only be valid for the whole of the universe. This means that such a determination of mass

would imply the observation of the movement of the center of the universe, which, Po-

incare concludes, is an absurdity.75

From his analysis of the processes of measurement of force and mass, he concludes that

it is impossible to give a satisfactory idea of mass and force within classical mechanics.76

3 Definitions in Textbooks

In a sample of about a hundred textbooks (Voigt 1901; Lenard 1936; Sommerfeld 1947;

Schaefer 1962; Budo 1974; Hestenes 1987; Alonso and Finn 1992; Daniel 1997; Gerthsen

2006; Kuypers 2008, among others), it was verified that ‘force is the cause of acceleration’

is the most common definition of force.

71 ‘‘l’acceleration de A n’est pas due seulement a l’action de B, mais a celle d’une foule d’autres corps C, D[…]’’ (1897, p. 735).72 ‘‘il faut donc decomposer l’acceleration de A en plusieurs composantes, et discerner quelle est celle deces composantes qui est due a l’action de B’’ (1897, p. 735).73 ‘‘Cette decomposition serait encore possible, si nous admettions que l’action de C sur A s’ajoute sim-plement a celle de B sur A, sans que la presence du corps C modifie l’action de B sur A, ou que la presencede B modifie l’action de C sur A; si nous admettions, par consequent, que deux corps quelconques s’attirent,que leur action mutuelle est dirigee suivant la droite qui les joint et ne depend que de leur distance; si nousadmettions, en un mot, l’hypothese des forces centrales’’ (1897, p. 735).74 ‘‘Mais avons-nous le droit d’admettre l’hypothese des forces centrales? Cette hypothese est-elle rigou-reusement exacte? Est-il certain qu’elle ne sera jamais contredite par l’experience? Qui oserait l’affirmer? Etsi nous devons abandonner cette hypothese, tout l’edifice si laborieusement eleve s’ecroulera. Nous n’avonsplus le droit de parler de la composante de l’acceleration de A qui est due a l’action de B. Nous n’avonsaucun moyen de la discerner de celle qui est due a l’action de C ou d’un autre corps. La regle pour la mesuredes masses devient inapplicable’’ (1897, p. 735–6).75 ‘‘Mais il n’existe pas de systeme soustrait a toute action exterieure; toutes les parties de l’Universsubissent plus ou moins fortement l’action de toutes les autres parties. La loi du mouvement du centre degravite n’est rigoureusement vraie que si on l’applique a l’Univers tout entier. Mais alors il faudrait, pour entirer les valeurs des masses, observer le mouvement du centre de gravite de l’Univers. L’absurdite de cetteconsequence est manifeste; nous ne connaissons que des mouvements relatifs; le mouvement du centre degravite de l’Univers restera pour nous une eternelle inconnue’’ (1897, p. 736).76 ‘‘nous devons conclure, qu’avec le systeme classique, il est impossible de donner de la force et de lamasse une idee satisfaisante’’ (1897, p. 736).

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Webster, for instance, wrote (1904): ‘‘The property of persistence thus defined is called

Inertia. This gives a criterion for finding whether a force is acting on a body or not […]

Force is acting on a body when its motion is not uniform’’ (p. 21).

In Feynman’s Lectures (1974), one reads: ‘‘The Second Law gave a specific way of

determining how the velocity changes under different influences called forces’’; and ‘‘If an

object is accelerating, some agency is at work’’ (§ 9–4).

Wolfson and Pasachoff, Physics (1990), write: ‘‘Why are we so interested in knowing

about forces? Because forces cause changes in motion’’ (p. 76).

If force is the cause of a motion which a body could not have by itself, force must be a

real existing thing. A contrary thesis has, however, been defended by some physicists.

Hamel in Mechanics (1912), says: ‘‘Force itself, however, we do not define as cause of

motion, force is a thing of thought and not a natural phenomenon’’.77

Platrier writes in Rational Mechanics, (1954): ‘‘In fact, force is only a human concept

and we have no knowledge of the profound cause of motions’’.78

Ludwig, in the Introduction to the Foundations of Theoretical Physics, (1985), defends

the thesis that the concept of force does not describe anything which exists in reality. In his

own terminology, force does not belong to ‘‘real text’’.79

Some physicists defend a variant of the most common definition of force, in under-

standing force as the effort felt by the pulling or pushing of an object. Planck (1916), for

instance, says that the cause of the motion is called force and ‘‘it corresponds to the effort,

which we feel, if that same motion had been produced through our muscles instead of the

bodies, which caused it’’.80

Nolting 2005, writes: ‘‘The concept of force can only be defined indirectly through its

effects. If we want to modify the state of movement or the shape of a body, for example,

using our muscles, then an effort will be necessary […] This effort is called force […] We

observe everywhere in our environment changes in the states of motion of certain bodies

[…] We see their causes equally in forces, which in the same way as our muscles, act on

the bodies’’.81

This kind of definition of force has its origin in Reech’s theory (1852). An important

follower of his was Jules Andrade, who wrote (1898), ‘‘Certain spirits despise the common

idea of force, as furthermore, they despise the notion of muscular force. This disdain does

77 ‘‘Die Kraft selbst aber definieren wir nicht als Ursache der Bewegung; denn die Kraft ist ein Gedank-ending und keine Naturerscheinung’’ (p. 56).78 ‘‘En realite la force ([F = m.a]) n’est qu’une conception humaine et la cause profonde des mouvementsnous est inconnue’’ (p. 112).79 ‘‘Der physikalische Begriff der Kraft beschreibt eben nicht etwas unmittelbar Feststellbares […] DerKraftbegriff gehort nicht zur Formulierung der Abbildungsprinzipien, die etwas im Realtext, d. h. an derWirklichkeit ([…]) Ablesbares in eine mathematische Form umzuschreiben gestatten’’ (p. 145).80 ‘‘Wir bezeichnen also nun ganz allgemein bei jeder beliebigen Bewegung die Ursache der Bewegung alsKraft und setzen ihre Große proportional der durch sie bewirkten Beschleunigung. Dieselbe entsprichtderjenigen Anstrengung, die wir verspuren wurden, wenn wir die namliche Bewegung, anstatt durch denbetreffenden Korper, durch unsere Muskeln hervorrufen wurden’’ (p. 10).81 ‘‘Der physikalische Begriff der Kraft laßt sich nur indirekt durch seine Wirkungen definieren. Wollen wirden Bewegungszustand oder die Gestalt eines Korpers z.B. durch Einsatz unserer Muskeln andern, so bedarfes einer Anstrengung, die um so großer ist, je großer die zeitliche Geschwindigkeitsanderung (Beschle-unigung) oder je starker die Deformation sein soll. Diese Anstrengung heißt Kraft. […] Nun beobachten wiruberall in unserer Umgebung Anderungen in den Bewegungszustanden gewisser Korper, ohne daß unsereMuskeln direkten Einfluß hatten. Ihre Ursache sehen wir ebenfalls in Kraften, welche in gleicher Weise wieunsere Muskeln auf die Korper einwirken’’ (p. 109).

104 R. L. Coelho

123

not seem justified to me, since the only common notion of force is the fruitful notion;

mechanics, we admit clearly, is essentially anthropomorphic’’.82

Poincare 1900, defended the thesis, however, that this notion of effort does not acquaint

us with the true nature of force.83 He adds, the anthropomorphism cannot provide the

foundation of anything truly scientific or philosophical.84

The fundamental equation of dynamics is sometimes used to define force. Fließbach

2007, for instance, writes as follows:

‘‘Newton’s second axiom embraces the following definitions and affirmations:

1. Definition of mass;

2. Definition of force

3. […]’’85

As the equation referred to as Newton’s second axiom (F = ma) is composed of three

variables, the definition of mass will have to be given by force and acceleration. As force is

defined by the same equation, it follows that it depends on what mass and acceleration are.

As, however, what mass might be depends on force, we remain not knowing what both are.

This kind of definition was criticized by Mach in 1868, as seen above (see Hestenes 1987,

p. 590; de Lozano and Cardenas 2002, p. 596).

Although the kinds of definitions of force considered above had already been criticized,

the criticism is rarely taken into account by modern authors.86 Let us consider if the

defining of force could be improved.

4 Philosophy

Newton’s force represents a deviation from the states referred to in the law of inertia. Euler

conceived force as a deviation from a certain motion in creating the theory of the motion

constrained by a surface. Reech criticized force because it was considered as a deviation

from a certain motion. In Hertz’s theory, there is force if there is a deviation from the

motion of the fundamental law. This concept, force as a deviation from a certain motion, is

also present in the decomposition of force.

A ‘deviation’ from a certain motion corresponds to the ‘negation’ of this motion, from a

logical point of view. If the motion has the properties ‘p and q’, the negation of this

conjunction is equivalent to the disjunction of the negations ‘non-p or non-q’. If it is

characterized as ‘rectilinear and uniform’, the negation is ‘non-rectilinear or non-uni-form’. The components of force are those which make the motion non-rectilinear or non-

uniform. These are therefore connected with the logical negation of the characteristics of

the motion of reference. If such a motion is characterized by ‘the shortest line and

82 ‘‘Certains esprits meprisent cette idee vulgaire de la force, comme ils meprisent d’ailleurs la notion del’effort musculaire. Ce mepris ne me paraıt pas justifie, car seule, la notion vulgaire de la force est la notionfeconde; la mecanique, avouons-le hautement, est essentiellement anthropomorphique’’ (p. 138).83 ‘‘cette notion d’effort ne nous fait pas connaıtre la veritable nature de la force’’ (1900, p. 468).84 ‘‘L’Anthropomorphisme a joue un role historique considerable dans la genese de la Mecanique; peut-etrefournira-t-il encore quelquefois un symbol qui paraıtra commode a quelques esprits; mais il ne peut reinfonder qui ait un caractere vraiment scientifique, ou un caractere vraiment philosophique’’ (1900, p. 468).85 ‘‘Das 2. Newtonsche Axiom beinhaltet folgende Definitionen und Aussagen: 1. Definition der Masse. 2.Definition der Kraft. 3. […]’’ (p. 13–14).86 French (1971, p. 170) is an exception to this: knowing the difficulties in defining force he does not giveany definition.

On the Concept of Force 105

123

uniformity’, the components of force are those which make the trajectory ‘non-the shortest’or the movement ‘non-uniform’. If the least curvature and uniformity characterize the

motion of reference, there is force if the curvature of the path is not a minimum or the

motion along it is not uniform. In this case, Hertz did not speak of components of force,

since he used coordinates for the decomposition.

With the introduction of coordinates, the information concerning the path and the

motion along it, is given through these. If m€xi ¼ 0 holds for each rectangular coordinate,

the motion is rectilinear and uniform. If not, it is non-rectilinear or non-uniform. This holds

mutatis mutandis for generalized coordinates. The example of the circular motion con-

sidered above shows: if mr2€h ¼ 0 the motion is uniform; if mr2€h ¼ Qh 6¼ 0; it is not. (We

say that Qh does not have the dimensions of force. This can now be understood in con-

nection with the motion of reference. In the former case, the motion of reference is the

rectilinear and uniform motion; in the latter, it is the circular uniform motion. As these

motions differ from each other, their variations in respect to time differ as well.)

This concept of force concerns a motion which is characterized by a trajectory and

how it is covered. This is not yet the concept of force we use when we say: ‘body A

exerts force f on body B’. It is true that the variable m appears in the equations referred to

above. However, mass does not matter, since those equations hold for all mechanical

bodies. This is however not the case concerning the left-hand side of F = ma, if F refers

to the force of a body. The force f also comes from experiments but not in the same way,

as we will see.

What can be drawn from experiments concerning this issue was presented in a clear way

by Helmholtz. In his Lectures on Theoretical Physics, (1911), he wrote: ‘‘Motion and

acceleration are facts, which can be observed […] On the contrary, if one speaks of force as

cause of this motion, one does not know more about the nature of force than can be

gathered from the observation of the occurrence of the motion […] Therefore, nothing can

be stated about force, which is not already known from acceleration’’.87 This can also be

shown in the following way.

Let us suppose that we have to study a motion about which we have no further infor-

mation. To study this motion, we have to observe it carefully. The best means of achieving

this goal are certainly stroboscopic images or filming. The result of this is some tens of

images. Thanks to them, we can measure the piece of the path in each interval of time,

determine the respective velocity and calculate the acceleration. This is all, however, we

can draw from the data. We can say nothing about the force or mass of the moving body

without further information. Acceleration is therefore the only one of the three magnitudes

which can be drawn from a phenomenon.

It follows that force requires more than one phenomenon. The proposition ‘A exerts

force f on B’ requires therefore more than one phenomenon. In order to assert that the force

of A is f, we have to carry out a set of experiments (Kohlrausch 1996, p. 133 ff; Arons

1990, p. 52 ff). In using f concerning body B, it is assumed that A exerts on B the ‘same

force’ as it has exerted in those experiments. Let us consider why it is said, that ‘A exerts

force’’ in all cases.

87 ‘‘Die Bewegungen und die Beschleunigungen sind Thatsachen, welche beobachtet werden konnen […]Wenn man dagegen von Kraften spricht als den Ursachen dieser Bewegungserscheinungen, so weiß manvon deren Wesen nichts weiter, als was man eben aus der Beobachtung des Bewegungsvorganges herau-slesen kann […] Man kann daher von der Kraft nichts aussagen, was man nicht bereits von derBeschleunigung weiss’’ (p. 24).

106 R. L. Coelho

123

As Bergmann and Schaefer highlighted, the only sign we have of force is acceleration.88

Between both acceleration and force there is, however, a necessary connection, which is

established by the law of inertia. The law states that a free body stays at rest or moves

rectilinearly and uniformly. Hence, ‘free body’ is a sufficient condition for constant

velocity. It follows from this implication that the moving body is not free if we observe

non-constant velocity. This reasoning (‘free body’ implies ‘constant velocity’ and ‘non-

constant velocity’ therefore ‘non-free body’) is logically correct. It is the modus tollens:

[(p ? q)^-q] ? -p. Thus, if the law of inertia is admitted, an accelerated motion

requires an external something, which causes its acceleration. Coherently, we say that body

A exerts force in all cases. In order to move on to ‘on body B’, we assume the ‘same force’.

Let us try to express this dealing with the phenomena without the theoretical constraint

derived from the law of inertia.

Thanks to a set of experiments, f has been ascribed to body A. Then we assume, the

body will move in the same way as it moved in those experiments. This assumption must

be made if we want to use the information drawn from those experiments. Introducing finto the equation F = ma, where ‘m’ and ‘a’ refer now to B, we can predict some results.

Thus, force f can be easily understood in conformity with our dealing with the

phenomena.

Let us turn now to the question asked in the introduction of how to conceive force in

compliance with the contributions of Newton, Euler, Lagrange, etc. and without the

inconveniences raised by the criticism of the concept. Thanks to Newton’s, Euler’s,

Lagrange’s or Hertz’s work we learn that force can be conceived as a deviation from a

certain motion. Taking each of these as a motion of reference and therefore force as a

deviation from the motion of reference, the criticism of concept can be overcome. It is not

necessary anymore to consider force as the cause of acceleration and to try to observe it.

Our dealing with phenomena is clarified by the meaning of force as a value, which is

ascribed to a body as a consequence of a certain set of experiments. All this enables us to

understand the problems with the concept.

D’Alembert’s and Carnot’s difficulties concern the lack of observability of force. What

they could observe were motions. Nevertheless, they admitted real forces. The admission

of what is not observable can be understood thanks to the law of inertia. Both accepted

this law as the first statement of their theories. Thus, if they had not admitted the

existence of force, their theories would not have been logically consistent. Tait, 1895,

expressed the relationship between the law of inertia and the definition of force in the

following clear way: ‘‘Thus, for the present, we have the definition of ‘‘force’’ as part of

this First Law: -Force is whatever changes the state of the rest or uniform motion of abody’’ (p. 5).

Even though the meaning of the law of inertia has changed, the structure of the law has

been maintained. Hence, its logical consequence is still the same: acceleration requires

force. As accelerated motions are observable, force must be there. The effects of force

being observable, it must be a real existing thing. Thus, the spreading of the definition

‘force is the cause of acceleration’ is understandable, since the law of inertia has been

accepted by almost all the authors (Coelho 2007).

88 ‘‘Das Tragheitsgesetz sagt aus, dass ein Korper weder eine positive noch eine negative Beschleunigungerfahrt, wenn keine außere Krafteinwirkung vorhanden ist. Beschleunigung ist also immer ein Anzeichen furdas Vorhandensein einer solchen außeren Einwirkung, und zwar das einzige, das die Mechanik kennt’’ (p.114).

On the Concept of Force 107

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Saint-Venant’s and Mach’s theories of mechanics start from acceleration. In a second

step, they define mass and finally force as a mere mathematical concept. That starting point

of their theories is, however, incompatible with the classical system, for acceleration

implies force, since the law of inertia is admitted. Hence, that sequence—acceleration,

mass, force—could be and was criticized for presupposing force before introducing it

(Roche 2006, p. 1030). This difficulty can now be overcome. If force is not anymore the

real cause of acceleration, a definition of mass based on its measurement, which involves

acceleration, is free of those objections (see Arons 1990, p. 51; Hecht 2006).

Kirchhoff (1897) defined mechanics as the science of motion and planned to carry out a

theory based on motion. He himself pointed out a difficulty: if there is a system of forces, it

is not possible to determine the components only through motion. If force is understood as

information drawn from some experiments, it follows that those components result from

previous experiments. As these experiments are motions, Kirchhoff’s plan is free of that

difficulty.

Hertz’s solution for force shows the difficulty in connecting force with phenomena. This

is corroborated by those authors like Hamel, Platrier or Ludwig, whose theses can be

summed up by ‘force is not in real text’. This kind of thesis is now clear: as we can observe

only acceleration, force cannot be seen in phenomena. As the determination of force is

based on observations of the mechanical kind, our knowledge of phenomena is limited by

the means employed in achieving it. Thus, it is understandable that Platrier 1954, said that

the ‘‘profound cause’’ of the motions is unknown to us. Wilczek 2004, says ‘‘By com-

parison to modern foundational physics, the culture of force is vaguely defined, limited in

scope, and approximate’’ (p. 12). In the next year, Wilczek characterizes the assumptions

concerning force as ‘‘a sort of folklore’’ (2005, p. 10).

The concept of force as a deviation from a certain motion enables us to integrate theses

of philosophers of science and to understand their criticism. As the real cause of accel-

eration is a theoretical consequence of the law of inertia, it is understandable what Russell

writes in Principles of Mathematics: ‘‘force is a mathematical fiction, not a physical entity’’

(1937, p. 482). Another mathematician and philosopher, Clifford wrote: ‘‘We do not know

why the presence of one body tends to change the velocity of another; to say that it arises

from the force resident in the first body acting upon the matter of the moving body is only

to slur over our ignorance’’ (1955, p. 243).

Nagel 1961, questions: ‘‘Why should uniform velocity be selected as the state of a body

which needs no explanation in terms of the operation of forces, rather than uniform rest or

uniform acceleration (such as motion along a circular orbit with constant velocity) […]?’’

(p. 177).

As force has been taken as a deviation from a motion of reference and the uniform

velocity is a characteristic of the motion of reference of the classical theory, there is no

place for force there. As from the standpoint of the classical theory, there is force if the

motion is not rectilinear, the circular motion implies force. We have, however, also seen

that the circular uniform motion can be taken as a motion of reference. In fact, this had

already been done by Lagrange’s formalism even though not verbalized. Hertz’s funda-

mental law also subsumes the circular motion. In sum, the ‘‘selection’’ referred to by Nagel

depends on the motion of reference. As this does not have to be the motion of the law of

inertia, the circular uniform motion can also be a motion of reference. Thus, the criticism is

overcome and the criticized aspects are integrated.

Ellis dealt with the concept of force in various papers (1962, 1963, 1965, 1976). His

thesis, the law of inertia lays down the mechanical behavior of bodies and force is a matter

of convention, was subject of intensive controversy (Hunt and Suchting 1969, pp. 235 ff).

108 R. L. Coelho

123

Let us consider this regarding the 1976 article: ‘‘it is a matter of convention what states we

should regard as natural, and hence what things we should regard as force-effects, and

hence what forces we should say exist’’ (p. 175).

The concept of force as a deviation from a certain motion enables us an easy under-

standing of the thesis on the conventionality, since a motion of reference implies some

choice. As it is not necessary for the motion of the law of inertia to be considered the

natural motion, and neither for the other motions with an analogous function, it is not

necessary to postulate the existence of forces anymore (Snider 1967).

In discussing the topic ‘causal laws’, Chalmers 2008, writes: ‘‘Newton’s laws can

readily be interpreted as causal laws describing the disposition of objects to exert and

respond to specified forces. However, this is not the only way […] The laws of mechanics

can also be written in a form that takes energy, rather than forces, as the starting point’’ (p.

223)’’. If force as the ‘cause of acceleration’ is a theoretical consequence of the law of

inertia and ‘force exerted’ by an object on another expresses a piece of information drawn

from other experiments, the Newtonian description of motion is not more ‘‘profound’’ than

the Lagrangian one.

5 Some Educational Implications

Students’ preconceptions or misconceptions and common sense beliefs concerning force

have been the subject of much research (McClelland 1985; Halloun and Hestenes 1985;

Bliss and Ogborn 1994; Hijs and Bosch 1995; Rowlands et al. 1999; de Lozano and

Cardenas 2002 among many others). Teaching strategies and methods have also been

developed (Arons 1990; Hestenes 1992; Rowlands et al. 1998; Stinner 2001; Galili 2001;

Seker and Welsh 2006). All this deserves special attention.89 In what follows, only a few

questions gathered from the literature will be dealt with.

The relationship between force and motion has been the subject of many investigations

and studies (Peters 1985; Halloun and Hestenes 1985; Galili and Bar 1992; Lombardi

1999; Carson and Rowlands 2005; Smith and Wittmann 2008). According to Rowlands

et al. 2007: ‘‘‘misconceptions’ of force and motion are fundamental because understanding

the Newtonian concept of force and motion is essential in understanding the system as a

whole’’ (p. 31). One typical issue of this problem concerns the relationship between force

and velocity, which is sometimes expressed as F = mv. This difficulty in learning could be

overcome thanks to the concept of force as deviation from a certain motion. In this case, to

speak of force already implies a motion. Without a motion of reference, force does not

have any meaning. The idea of ‘‘deviation’’ from the motion of the law of inertia is used by

Hestenes’ New Foundations for Classical Mechanics (1987, p. 589) (see also Arons 1990,

p. 52).

Carson and Rowlands 2005, write: ‘‘The problem is that we do not observe or expe-

rience ‘force’ as such’’ (p. 474). It is hence understandable that ‘‘it is difficult to see how

force can be abstracted from experience’’ (p. 479). The forceless situation could be helpful

in enabling us to make a comparison with situations with force. This situation is however

unreal (Carson and Rowlands 2005, p. 483, 486–7). Let us consider if the difficulty could

be overcome.

89 These issues deserve special attention because of the considerable amount of research literature. Someempirical educational research is also in preparation within the framework of the European Project Historyand Philosophy in Science Teaching.

On the Concept of Force 109

123

Let us take the linear oscillation of a spring as an example. It can be verified through

observation that a body and a spring are involved in this motion. The motion of the body is

accelerated as well as each element of the spring. If it is said that the spring exerts force fon the body and force is the cause of acceleration, we are led to looking for the cause of the

acceleration there. However, it is difficult, perhaps impossible, to distinguish between

cause and effect there, since both bodies are involved in the motion. If it is said that f was

drawn from other experiments, it is not necessary to ‘‘see’’ force there. In general, if it is

taught that force is there, where the motion is accelerated, a student will try to find in

motion and through the observation of it, what does not come from there. If it is taught that

force were gained from other experiments, the student will understand it without difficulty.

In a study on scientific argumentation in the classroom, Driver et al. 2000, write: ‘‘As

Kress, Ogborn, Jewitt, and Tsatsarleis (1998) pointed out nature does not ‘‘speak for

itself,’’ particularly when the teacher is trying to convince pupils […] that objects continue

in motion forever’’ (p. 291). Hanson 1965, highlighted the logical component of this

problem. A frame of reference requires four particles. It is admitted in physics that any two

particles attract each other. Thus it is impossible to determine how a ‘‘free body’’ moves (p.

14). Matthews 2008, put forward a radical question: ‘‘we never see force-free behaviour in

nature, nor can it be experimentally induced, so what is the source and justification of our

knowledge of bodies without impressed forces?’’ (p. 10)

According to textbooks on mechanics, the law of inertia comes from Newton. It is perhaps

‘‘difficult’’ to accept a change of the status of a motion which has been adopted since the

seventeenth century. However, a change in the law has already taken place, as we will see.

According to Newton (1726), d’Alembert (1758), Laplace (1799), Carnot (1803),

Poisson (1833) and many others, a ball on a flat table justifies the law of inertia. In fact, it

can be observed that it stays at rest or moves rectilinearly and uniformly if it is not

disturbed by an impressed force. The difficulty at that time was the uniformity, which

could not be observed. For this reason, the staying at rest and moving rectilinearly were

laws of nature in d’Alembert’s theory and the uniformity of motion was a corollary, as it

could not be observed but only inferred. In contemporary mechanics, a ball on a flat table

cannot however be used for the same aim. It is not a free body and the law is formulated for

such a one. Nowadays, ‘‘free body’’ means a body without any constraint, whereas in the

past, the body was free only in some directions and not in all thinkable ones. The meaning

of ‘‘free body’’ differs, therefore, from ‘‘body not disturbed by impressed force’’. For this

reason, it is now impossible to outline an experiment in compliance with the law, whereas

the law was proved by experiments in the past. If we adopt contemporary statements of the

law, we will have some problems, as has been pointed out.

Considering the motion of the law of inertia as a motion of reference, the law of the past

is integrated in its experimental component. Since the contemporary meaning of ‘‘free

body’’ is avoided, it is not necessary to prove what cannot be. Furthermore, other state-

ments with analogous function, which appeared in the course of the development of

mechanics, such as Hertz’s fundamental law, can be integrated as well. In so far as the

ascribing of a natural motion to bodies has led to the concept of force as a real something,

the introducing of the motion of reference avoids the traditional problem with force.

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Author Biography

Ricardo Lopes Coelho has been a ‘‘Professor Auxiliar’’ at the Faculty of Sciences of the University ofLisbon, since 1997, and a ‘‘Privatdozent’’ at the Technical University of Berlin, since 2001. He studiedpiano, philosophy and physics in Portugal, did his PhD at the TU-Berlin and his Habilitation in History andPhilosophy of Exact Sciences at the same University. Among others, he published some articles concerninghis main research interest, the understanding of scientific concepts and principles through its past andphilosophy. He is the author of two books: Zur Konzeption der Kraft der Mechanik (On the Concept ofForce in Mechanics) (2001) and O Conceito de Energia: Passado e Sentido (On the Concept of Energy:History and Meaning) (2006).

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