The mass of a body is its inertia or resistance to change of motion. More precisely, it is a property of the body that determines the body's acceleration under the influence of a given force. Mass can therefore be measured either by the amount of force necessary to impart to the body a given motion in a given time or by the acceleration produced by a given force.
The absolute metric unit of mass is the gram, which is the mass of a body whose velocity increases by one centimeter per second each second if acted upon by a force of one dyne. Other common units are the kilogram (1,000 grams) and the pound (453.592 grams). For velocities that are small as compared with the speed of light, the mass of a body is a constant, characteristic of the body and independent of its location—in contrast to weight, which varies with the body's place on Earth or in the universe.
Although fundamental to science and, together with length and time, the basis of all measurements in physics, the concept of mass was unambiguously defined only at the end of the nineteenth century. However, its rudimentary sources, systematically employed long before by Isaac Newton and to some extent already by Johannes Kepler, can be traced back to early Neoplatonic ideas concerning the inactivity of matter as opposed to the spontaneity of mind. The ancient metaphysical antithesis of matter and spirit served as a prototype of the physical contrast of mass and force.
Concept of Inertial Mass
Antiquity, and Greek science in particular, had no conception of inertial mass. Even the idea of quantity of matter (quantitas materiae ), the antecedent of inertial or dynamic mass, was foreign to the conceptual scheme of Aristotelian natural philosophy. Paradoxically, it was Neoplatonism and its admixtures of Judeo-Christian doctrines, with their emphasis on the spiritual and immaterial nature of reality, that laid the foundations for the inertial conception of mass, which later became the basic notion of materialistic or substantial philosophy. To accentuate the immaterial, sublime source of all force and life in the intellect or God, Neoplatonism degraded matter to impotence and endowed it with inertia in the sense of an absolute absence of spontaneous activity. For Plotinus, Proclus, Philo, Ibn Gabirol, and the Platonic patristic authors, matter was something base, inert, shapeless and "plump," attributes that reappear in Kepler's characterization of matter as that which is too "plump and clumsy to move itself from one place to another."
The idea of a quantitative determination of matter different from, and ontologically prior to, spatial extension originated in scholastic philosophy in connection with the problem of the transubstantiation. The question of how accidents of condensation or rarefaction (volume changes) can persist in the consecrated hostia of the holy bread and wine of the Eucharist whereas the substances of the bread and the wine change into the Body and the Blood of Christ led Aegidius Romanus, a disciple of Thomas Aquinas, to the formulation of his theory of duplex quantitas. According to this theory matter is determined by two quantities; it is "so and so much" (tanta et tanta ) and "occupies such and such a volume" (et occupat tantum et tantum locum ), the former determination, the quantitas materiae, having ontological priority over bulk. Aegidius's early conception of mass as quantity of matter, expounded in his Theoremata de Corpore Christi (1276), was soon renounced and had little influence on the subsequent development of the concept of mass. It was primarily Kepler who ascribed to matter an inherent propensity for inertia in his search for a dynamical explanation of the newly discovered elliptical orbits of planetary motion; in need of a concept expressing the opposition intrinsic in matter to motory forces, Kepler formulated the inertial concept of mass. In his Epitome Astronomiae Copernicanae (1618) he declared that "inertia or opposition to motion is a characteristic of matter; it is stronger the greater the quantity of matter in a given volume."
A different approach to the same idea arose from the study of terrestrial gravitation. As soon as gravity was regarded no longer as a factor residing in the heavy body itself, as Aristotle taught, but as an interaction between an active principle, extraneous to the gravitating body, and a passive principle, inherent in matter, as Alfonso Borelli and Giovanni Baliani (author of De Motu Gravium, 1638) contended, the notion of inertial mass became a necessity for a dynamical explanation of free fall and other gravitational phenomena. Furthermore, Christian Huygens's investigations of centrifugal forces (De Vi Centrifuga, 1659; published in Leiden, 1703) made it clear that a quantitative determination of such forces is possible only if with each body is associated a certain characteristic property proportional to, but conceptually different from, the body's weight. Finally, the systematic study of impact phenomena, carried out by John Wallis, Sir Christopher Wren, and Huygens, enforced the introduction of inertial mass. With Newton's foundations of dynamics (Principia, 1687) these four categories of apparently disparate phenomena (planetary motion, free fall, centrifugal force, and impact phenomena) found their logical unification, through his consistent employment of the notion of inertial mass. Newton's explicit definition of this concept, however, as "the measure of quantity of matter, arising from its density and bulk conjointly" was still unsatisfactory from both the logical and the methodological points of view. It was probably the influence of Kepler or of Robert Boyle and his famous experiments on the compressibility of air that made Newton choose the notion of density as a primary concept in his peculiar formulation of the definition of mass, a formulation that was severely criticized in modern times, especially by Ernst Mach and Paul Volkmann.
Leibniz and Kant
Gottfried Wilhelm Leibniz's original conception of mass (1669), in contrast to Newton's, defined it as that property which endows primary matter with spatial extension and antitypy, or impenetrability. In his later writings, especially in his doctrine of monads, Leibniz associated mass with secondary matter and saw in it a property of a collection of substances (monads) resulting from their being a collection. Finally, recognizing the insufficiency of purely geometric conceptions to account for the physical behavior of interacting bodies, Leibniz departed from the Cartesian approach and accepted the dynamic, or inertial, conception of mass. The trend of Leibniz's ideas was brought to its final consequences by Immanuel Kant, with his rejection of the Newtonian vis inertiae, the dynamic opposition against impressed force. Refuting its legitimacy on the ground that "only motion, but not rest, can oppose motion," Kant postulated the law of inertia as corresponding to the category of causality ("every change of the state of motion has an external cause") and consequently defined mass as the amount of the mobile (die Menge des Beweglichen ) in a given volume, measured by the quantity of motion (Die metaphysischen Anfangsgründe der Naturwissenschaft, 1786).
Definition of Mass
Under the influence of the Kantian formulation, often incompletely understood, and primarily owing to the fact that in spite of the universal use of the concept in science as well as in philosophy no clear-cut definition of mass was available, most authors defined mass as quantity of matter without specifying how to measure it. Toward the middle of the nineteenth century, with the rise of modern foundational research and the critical study of the principles of mechanics, the logical deficiency of such definitions became obvious. It was primarily Ernst Mach, preceded by Barré de Saint-Venant and Jules Andrade, who insisted on the necessity of a clear operational definition of mass. In an essay, "Über die Definition der Masse" (1867; published in 1868 in Carl's Repertorium der Experimentalphysik, Vol. 4, pp. 355–359), and in the Science of Mechanics (Die Mechanik in ihrer Entwicklung, historisch-kritisch dargestellt, Leipzig, 1883; translated by T. J. McCormack, La Salle, IL, 1942), Mach defined the ratio of the masses of two bodies that interact with each other but are otherwise unaffected by all other bodies in the universe as the inverse ratio of their respective accelerations (m 1/m 2 = a 2/a 1), thereby converting Newton's third law of action and reaction to a definition of mass. If a particular body is chosen as the standard unit of mass, the mass of any other body can be unambiguously determined by simple physical operations. The practical method of comparing masses by weighing is, of course, operationally still simpler but logically more complicated, since the notion of weight presupposes that of mass. Although Mach's definition is not quite unobjectionable, it has gained great popularity and is generally adopted in modern texts in science.
Inertial and Gravitational Mass
In addition to its inertial mass, every physical body possesses gravitational mass, which determines, in its active aspect, the strength of the gravitational field produced by the body and, in its passive aspect, the amount by which the body is affected by the gravitational field produced by other bodies. According to Newton's law of universal gravitation, the force of attraction is proportional to the inertial masses of both the attracting and the attracted bodies. The resulting proportionality of inertial and gravitational masses of one and the same body, experimentally confirmed by Newton, Friedrich Bessel, Roland von Eötvös, and others, remained in classical physics a purely empirical and accidental feature, whereas the strict proportionality between the active and the passive gravitational masses is a straightforward consequence of Newton's third law of action and reaction or, alternatively, of the very definition of inertial mass if the postulated interaction is of gravitational nature. In general relativity, however, the so-called principle of equivalence, which maintains the unrestricted equivalence between uniformly accelerated reference systems and homogeneous gravitational fields, implies the fundamental identity between inertial and passive gravitational masses. In addition, it can be shown that on the basis of general relativity the active gravitational mass of a body or dynamical system equals its inertial mass, so that in relativistic physics, in contrast to Newtonian physics, the identity of all three kinds of masses is a necessary consequence of its fundamental assumptions.
Mass and Energy
Whereas general relativity led to an important unification of the concept of mass, special relativity already, with Albert Einstein's paper Does the Inertia of a Body Depend upon Its Energy Content? (1905; reprinted in The Principle of Relativity, New York, 1923), led to a vast generalization of the concept by showing the equivalence of mass and energy insofar as a body emitting radiative energy of an amount E loses mass to an amount of E/c 2, where c is the velocity of light. Subsequent research, especially in connection with energy transformations in nuclear physics, supported the general validity of the formula E = mc 2, according to which mass and energy are interconvertible and one gram of mass yields 9×1020 ergs of energy. It also became obvious that Antoine Lavoisier's law of the conservation of mass (1789) and Robert Mayer's (or Hermann Helmholtz's) law of the conservation of energy were only approximately correct and that it was the sum total of mass and energy that was conserved in any physicochemical process.
Influence of the Electromagnetic Concept
The way to these far-reaching conclusions of relativity had been prepared to some extent already by the introduction of the electromagnetic concept of mass at the end of the nineteenth century (by J. J. Thomson, Oliver Heaviside, and Max Abraham). It seemed possible on the basis of James Clerk Maxwell's electromagnetic theory to account for the inertial behavior of moving charged particles in terms of induction effects of purely electromagnetic nature. Walter Kaufmann's experiments (1902) on the deflection of electrons by simultaneous electric and magnetic fields and his determination of the slightly variable inertial mass of the electron seemed at the time to support the hypothesis that the mass of the electron, and ultimately the mass of every elementary particle, is of purely electromagnetic nature. Although such eminent theoreticians as H. A. Lorentz, Wilhelm Wien, and Henri Poincaré accepted these ideas, according to which the whole universe of physics is but an interplay of convection currents and their radiation, with physical reality stripped of all material substantiality, the electromagnetic conception of mass had to make way for the relativistic concept as outlined above. Certain aspects of the electromagnetic conception of mass did survive, however, and reappeared in modern field theories—in particular the fundamental tenet that matter does not do what it does because it is what it is, but it is what it is because it does what it does.
See also Aristotle; Boyle, Robert; Energy; Ibn Gabirol, Solomon ben Judah; Kant, Immanuel; Kepler, Johannes; Leibniz, Gottfried Wilhelm; Mach, Ernst; Maxwell, James Clerk; Neoplatonism; Newton, Isaac; Patristic Philosophy; Philo Judaeus; Plotinus; Poincaré, Jules Henri; Proclus; Thomas Aquinas, St.
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