The Greek word “maza” originally meant an inferior quality of bread.[1] Archimedes used the related word “masse” to mean a lump or block. The Latin word of the early Christian Church, “massa,” expanded the meaning to include an “aggregation of bodies.”[2] In the writings of Aristotle, the word “materia” originally meant timber.[3] Then, in the 13th century, a disciple of Thomas Aquinas (Aegidius Romanus) conceived the idea of quantitas materiae (“quantity of matter”) “as a measure of mass or matter, independent of determinations of volume or weight.”[4] Throughout most of recorded history, matter and mass were thought of as synonymous terms. For example, Galileo considered “mass” as “another name for matter itself.”[5]

Kepler, Galileo, Descartes, and Newton, working separately, created the concept of inertia, which in concert with Newton’s second law of motion would ultimately result in the concept of “inertial mass.” Kepler described inertia (the resistance of a body to being moved) as “something similar to weight” which “matter must have.” Kepler also stated that the “repugnance [resistance of] inertia or opposition to motion is a characteristic of matter; it is stronger, the greater the quantity of matter in a given volume.”[6] Newton equated a body of mass (corporis vel massae) with its quantity of matter (quantitas materiae), however Newton’s three laws of motion made no explicit mention of mass. The concept of mass is only implied by the acceleration of a body in his second law.[7] Thus, it was left for Euler in 1736 to explicitly state a formula for “inertial mass:” “Force equals mass times acceleration.”[8] Reciprocally, m = F/a.

During the early part of the 18th century, German scientist Gottfried Leibniz (1646 – 1716) invented complicated and confusing theories of mass and matter. In the mid 18th century, Immanuel Kant criticized Newton’s concepts, but failed to advance the concept of mass. The idea of the indestructibility and “conservation” of matter or mass was implicit in Newton’s theories, and French physicist Antoine Lavoisier (in 1789) extended this notion of conservation to chemical reactions,[9] but he failed to define either matter or mass.[10] During the 19th century, French physicist Bané de Saint-Venont, Maxwell, Ernst Mach and Hertz all attempted to invent new definitions of mass. Mach and Saint-Venont each rejected quantitas materiae, but failed to come up with an improved alternative, although Mach did arrive at a theoretical construct.[11] Maxwell’s theory assigned a priority of force over mass, and Hertz’s definition essentially redefined mass by its weight.[12]

During the early 20th century, Italian scientists contrived numerous definitions of mass but without much success. Their attempts only resulted in several definitions of “gravitational mass.”[13] Later, scientists Hans Herm (in 1938), Pendse (in 1939), Herbert Simon (in 1947), and Alfred Tarshi attempted to define the concept of mass as a “primitive” concept, without using other concepts to make it meaningful.[14] Two of such attempts merely resulted in mass ratios that can be obtained by Euler’s formula, and the others attempted to define mass by statistical inference. All involved the motion of bodies by reason of force.

Jammer concluded that in general, “[N]o attempts to formalize Newtonian mechanics by a precise explicit definition of mass have been very successful,” and he gave as an example the following circular definitions:

“We obtain our knowledge of forces by having some theory about masses, and our knowledge about masses by having some theory about forces.”[15]

Thus, it may be impossible to devise a meaningful qualitative definition of mass without resort to an empirical and quantitative interaction between the variables of Euler’s formula: m = F/a. As one sage writer concluded, “Mass is a mess.”