304 The Physics Teacher ◆ Vol. 50, May 2012 DOI: 10.1119/1.3703550

Newton uses the terms “body” or “mass.” In his book, the word “body” was used more often than “mass.”6

Ten years after the third and last edition of Newton’s Prin-cipia, Euler published the first book on analytical mechanics. In this book, Mechanics or the Science of Motion Presented Analytically (1736), a new definition of mass, a dynamic one, appeared. The mass of a body is considered proportional to the number of points that compound (comprise) it. Two such points are equal only if the effect of any force upon each point is the same. The quantity of matter was then redefined: “Concerning the quantity of matter, two bodies which are compounded by the same number of points are equal to each other.” 7

In 18th-century mechanics, the terms “body,” “mass,” and “quantity of matter” appeared as synonymous. D’Alembert, for instance, says in his famous Treatise of Dynamics that the quantity of motion is the product of mass and velocity, but he also speaks of the product of body and velocity.8 Toward the end of this century and in the following, the two definitions of mass—the Newtonian and Eulerian—were both common. Poisson gives us an example of dealing with the two connec-tions of the concept of mass: with quantity of mass and force. At the beginning of his Treatise on Mechanics (1833), mass is defined as quantity of matter. Later on, for measurements, he uses the equation F=ma.9 There was no criticism of this. Indeed, the difficulty in defining mass only emerged when physicists tried to find a solution for the problem of force based on mass.

The problem with the concept of force had been pointed out by d’Alembert (1743) and Carnot (1803), among others. These physicists, however, did not only criticize the concept of force but also developed new theories of mechanics in or-der to avoid this concept.10 The critical point was that force was understood as the cause of acceleration and this cause was in general not observable. As they did not want to base mechanics on what is not observable, they developed new theories. Toward the middle of the 19th century, another French physicist, Barré de Saint-Venant, made the first at-tempt of solving the problem of force by means of mass.

Barré’s Principles of Mechanics (1851) starts with one unique axiom. This states that acceleration is caused by ma-terial points in reciprocal action. Based on this, he defines mass: two points have equal mass if they give each other the same opposite velocity, in consequence of an impact to which they had the same and opposite velocity.11 Finally, he defines force as a mere product of these two magnitudes, mass and acceleration. In sum, in order to solve the problem of force, Barré proposed a new sequence for the terms of the funda-mental equation of dynamics: first acceleration, second mass, third force.

On the Definition of Mass inMechanics: Why Is It So Difficult?Ricardo Lopes Coelho, University of Lisbon, Lisbon, Portugal

In spite of the concerted efforts of physicists, philoso-phers, mathematicians, and logicians, no final clarifica-tion of the concept of mass has been reached. So con-

cludes Jammer in his book on the history of the concept.1 The Nobel laureate Wilczek called our attention to the problem in his papers on the concepts of the fundamental equation of dynamics.2 In 2005, Roche wrote a paper whose title asks the question “What is mass?”3 Hecht sums up the situation in textbooks in the title of his article “There Is No Really Good Definition of Mass.”4 Where the difficulty in defining mass in classical mechanics lies is the question addressed in the present paper. Before dealing with this topic, it is important to make explicit what the problem consists of, since sometimes people think that there is no problem with mass at all.

It is very common to find definitions of mass that are pro-posed for the only reason that they are in complete agreement with a certain measurement process. Indeed, the difficulty with the definition of mass does not consist of verbalizing a process of measurement but rather in obtaining a concept that fits the whole of mechanics and is logically permissible. There is, for instance, no objection to define mass by means of the fundamental equation of dynamics: mass is the quotient of force and acceleration. If the same equation is also used to define force, then an objection emerges since in such a case the definition is logically vicious.5 Indeed, if mass is defined by force and force by mass, we will have at the end no knowledge of what either force or mass is. Many attempts have been made to define mass. The main trends in defining it have their his-torical origin in the past centuries.

Historical topicsThis historical overview aims not to present what happened

but rather to show how the current situation came about. The reader will easily find parallels between contemporary defini-tions of mass and concepts of 19th-century physics or earlier. This is also useful in physics teaching since sometimes stu-dents come up with ideas that are not entirely acceptable but remind us of concepts of the past. If the connection with the past is made, students will try by themselves to change their way of thinking about the concepts.

Let us start with Newton’s Mathematical Principles of Natu-ral Philosophy (1687). The foundations of his theory consist of eight definitions and three axioms. The key concepts of the definitions are: matter, motion, and force. Five of the eight propositions define quantities and the other three define con-cepts. Matter and motion are not defined, but their quantities are. The quantity of matter, the first of the definitions, is pre-sented as the product of density and volume. For this quantity,

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of force as cause of acceleration.14 In order to achieve his goal, Kirchhoff established that the task of mechanics is not to ex-plain phenomena but only to describe motion in the simplest way. He took then space, time, and matter as the primitive no-tions of his theory. Force and mass became mere theoretical concepts.15

In Hertz’s Principles of Mechanics (1894), a peculiar solu-tion for mass and force was proposed. Mass and force are con-cepts created by us, according to Hertz. They are consequent-ly a priori, i.e., independent of experience. In a second step, Hertz establishes a correspondence between these concepts and the measurements of the magnitudes mass and force. Thus, when we speak of mass, the reference of the concept is a process of measurement and vice versa; when we measure the mass of a body, we have to connect this with the concept of mass. The same holds for force. These are perhaps the most abstract concepts of mass and force proposed until then.

In 20th-century textbooks, the definition of mass as quantity of matter still appears, but it is rare and has been criticized. Instead of this, mass has been defined dynami-cally.16 This definition has appeared in two forms, which can be characterized by the sequences of the concepts. When it is said that force f acts on a body of mass m and produces accel-eration a, mass appears in the sequence “f, m, a.” In this case, the definition of mass depends on force. As the concept of force is a problem on its own, this sequence will not be dealt

Mach’s definition of mass was proposed for the first time in a short paper in 1868. Mach also faced the problem of force. He said physicists defined mass through weight (m=W/g) and weight through mass (W=mg). To avoid this vicious circle, he proposed the following sequence: first, to consider reciprocal acceleration between bodies as a matter of fact; second, to define mass through impact between bod-ies; and third, to define force as the product of mass and ac-celeration.12 This sequence—first acceleration, second mass, third force—is similar to Barré’s and will be taken up again by Mach in his Mechanics.

There was a group of physical engineers whose subject was technical mechanics, who do not seem to have had any problem with the concept of mass. Poncelet, e.g., in his book Introduction to Industrial Mechanics (1870)13 defines the ki-netic energy of a body as Pv2/(2g), where P stands for weight, g for the local free-fall acceleration, and v for the velocity of the body. This is not only a mere formality, since the weight P of a body and the local acceleration g could be measured. The term mass is introduced later on in the course of the book but only as an abbreviation of P/g.

The German physicists Gustav Kirchhoff and Heinrich Hertz presented a new approach to the problems of force and mass. In Kirchhoff ’s Lectures on Mechanics (1876), the basis for a reorganization of mechanics is presented. This reorga-nizing was needed, according to Kirchhoff, due to the concept

306 The Physics Teacher ◆ Vol. 50, May 2012

but has the disadvantage of contradicting the interpretation of F=ma, which is logically compatible with the law of inertia. The real difficulty with that definition of mass lies, therefore, in Newton’s first law of motion.

References1. M. Jammer, Concepts of Mass: In Classical and Modern Physics

(Dover, NY, 1997), p. 224.2. F. Wilczek, “Whence the force of F = ma? I: Culture shock,”

Phys. Today 57 (10), 11–12 (2004); F. Wilczek, “Whence the force of F =ma? III: Cultural diversity,” Phys. Today 58 (7), 10–11 (2005).

3. J. Roche, “What is mass?” Eur. J. Phys. 26, 225–242 (2005).4. E. Hecht, “There is no really good definition of mass,” Phys.

Teach. 44, 40–45 (Jan. 2006).5. See T. Fließbach, Lehrbuch zur theoretischen Mechanik.

Mechanik, 5th ed. (Spektrum Akademischer Verlag, Heidel-berg, Berlin, Oxford, 2007), Vol. I, p. 13-4.

6. F. Steinle, “Was ist Masse? Newtons Begriff der Materiemenge,“ Philosophia Naturalis 29, 95 (1992).

7. L. Euler, Mechanica sive motus scientia analytice exposita Vol. I (Saint-Pétersburg, 1736 or Opera Omnia, Ser. II, Vol. 1) § 140.

8. J. d’Alembert, Traité de Dynamique (Paris, 1758, and Rep. Johnson Reprint Corporation, New York) § 51.

9. S. D. Poisson, Traité de Mécanique (Bachelier, Paris, 1833) p. 1, 223.10. See M. Jammer, Concepts of Force (Dover, NY, 1999), pp. 212–

215; R. Coelho, “On the concept of force: How understanding its history can improve physics teaching,” Sci. Educ. 19, 94–97 (2010).

11. A. J. C. B. de Saint-Venant, Principes de Mécaniques fondés sur la Cinématique (Bachelier, Paris, 1851) § 81.12. E. Mach, “Ueber die definition der masse,“ Repertorium für

Experimental-Physik 4, 356 (1868).13. V. Poncelet, Introduction à la Mécanique Industrielle (Gauthier-

Villars, Paris, 1870), p. 121.14. This was the great problem in theoretical physics at that time

since it concerns the foundations of mechanics and mechanics constituted the basis of the whole of physics. His book was very successful. Its second edition appeared in the same year.

15. G. Kirchhoff, Vorlesungen über Mathematische Physik, 4th ed. (Teubner, Leipzig, 1897), Vol. I, p. 1.

16. See L. Eisenbud “On the classical laws of motion,” Am. J. Phys.26, 144–158 (1958); R. Weinstock “Laws of classical motion. What’s F? What’s m? What’s a?” Am. J. Phys. 29, 698–702 (1961); Ref. 4; C. Kalman “On the concept of force: A comment on Lopes Coelho,” Sci. Educ. 20, 67–69 (2011).

17. E. Mach, The Science of Mechanics: A Critical and Historical Ac-count of its Development, translated by T. J. McCormack (The Open Court Publishing Company, Chicago, 1902), p. 218.

18. See R. Coelho, “Conceptual problems in the foundations of mechanics,” Sci. Educ.; online first www.springerlink.com/content/b0m6258371775000/fulltext.pdf (2011).

Ricardo Lopes Coelho has been an assistant professor at the Faculty of Sciences of the University of Lisbon and a Privatdozent at the Technical University of Berlin. His main research interest is conceptual problems in the foundations of physics. rjcoelho@fc.ul.pt

with in what follows. The other way of defining mass, first proposed by Saint-Venant and independently by Mach, will be considered.

The problem with Mach’s definitionAccording to Barré and Mach, the mass of a body can be

determined by means of an elastic impact of two bodies. Let us take this example. Due to an elastic impact of two bodies, there is a change in the velocities of both bodies. Taking the mass of one of them as a unit, the mass of the other can be de-termined by the inverse of their accelerations, i.e.

17

Using the reciprocal accelerations of the two bodies to define mass, Mach defines force as a mere product of acceleration and mass.

Mechanics teaches us, however, that acceleration is caused by force. This has as a consequence that force is prior to accel-eration. In this case, it does not make sense to start a sequence of the concepts by acceleration and end it with force, as pro-posed by Saint-Venant and Mach. Let us return to the point that prevents us from using this sequence and ask the ques-tion of why we say that force causes acceleration.

The law of inertia states that a free body (no net force act-ing) maintains its state of resting or of moving rectilinearly and uniformly. This means that “free body” is a sufficient condition for constant velocity. Let us write this in the form:

free body ➯ constant velocity.

It follows from this implication that, if the velocity is not con-stant, the body is not free, i.e.,

non-constant velocity ➯ non-free body.

Non-constant velocity means acceleration. Therefore, accel-eration is a sufficient condition for a non-free body.

If a body is not free, then something must be acting on it. That something that acts on a body is called “force” in mechanics. Therefore, force is the necessary condition for acceleration. If force is the necessary condition for accelera-tion, it is not logically acceptable to start with acceleration. This starting point is, however, required by any definition, à la Mach.

This logical approach to the law of inertia makes under-standable, on one hand, that most physicists have defended that force is the cause of acceleration, despite the criticism of this concept. Indeed, this concept of force is completely consistent with the first law of mechanics. On the other hand, this approach shows why Barré’s and Mach’s claim could not have been accepted by mechanics. Their claim contradicts a logical consequence of the law of inertia.18

In conclusion, a definition of mass à la Mach has the ad-vantage of giving mass by means of a measurement process,