The pre-Newtonian meaning of the word “weight”; a comment on “Kepler and the origins of pre-Newtonian mass” [Am. J. Phys. 85, 115–123 (2017)] Free

Eugene Hecht's recent scholarly work¹ traces the concept of “mass” from medieval times to Newton's Principia,² and adds much to our understanding of pre-Newtonian mass, and in particular, the gradual change from philosophical to scientific thinking. With the possible exception of his three laws of planetary motion, Kepler's contribution is often neglected, but it had a significant effect on Newton's discoveries. Astronomer-cosmologist Carl Sagan³ even suggested that without Kepler “Newtonian Physics might not have come to be.” It is no wonder that Newton⁴ said “If I have seen further, it is by standing on the shoulders of giants” of whom Kepler was certainly one.

While Newton defined mass by his second law as m = f / a (measured in mass units) and weight as w = mg (measured in force units) Hecht discussed pre-Newtonian mass only, so it is suggested here that another important concept, which Hecht did not deal with, is the pre-Newtonian meaning of the word “weight,” because surprisingly, it was not a measure of weight as it is now known, but it was actually the measure of mass in pre-Newtonian times. It is not always realized that Newton actually changed the language of physics in this respect.⁵ 

Ever since the beam balance was invented over three millennia ago, it has been used to measure what we now know as mass, which, until the time of Newton, was called “weight,” e.g., a “weight of 10 pounds.”⁶ However, the ancients believed that the beam balance also measured the associated downward force (of gravity) or “heaviness” which they recognized through their muscular senses. It does not.⁷ Nor, until the time of Hooke⁸ and Newton, was there any known method of measuring what we now call weight or force of gravity on the mass. Nevertheless, mass and its associated downward force were believed to be one and the same thing which was called “weight.”⁹ Balance is achieved by equalizing (but not measuring) the gravitational forces acting on each side by adjusting standard “weights” on one side (which post-Newton are properly called masses) but the beam balance is quite incapable of measuring the actual gravitational forces. This misconception appears to be one reason why Max Jammer¹⁰ came to the conclusion that “the ancients did not form a concept of mass.” In fact they did, but called it “weight.” Thus, pre-Newton “weight” had a dual meaning, weight as mass and weight as gravitational force.

On the one hand, the purveyors of all types of goods, their customers and the statutory regulators all believed that “weight” (as measured by a beam balance) was a measure of the quantity of a material. The primary purpose of weight standards was to promote fair trade, and in this respect weight was related to the quantity of goods, which we now call mass.¹¹ A goldsmith and his customer, for example, both believed that an ounce of gold was a quantity of gold, not a downward force which had absolutely no relevance to their transaction. The major question here is whether a pound of lead is the same “quantity of matter” as a pound of beans, or a pound of sand. These all balance one against the other but do they contain the same amount of matter? In engineering, 1000 pounds of wood inside a building has the same effect on the foundations as 1000 pounds of bricks. On a ship, they would have the same effect on the depth of water relative to the hull. So, in engineering terms, and at least in these cases, both could be regarded as the same quantity of matter producing the same downward force.

On the other hand, Clagett¹² says:

Archimedes,¹³ for example, expressed his principle in terms of “weight” although he could not measure it (as a force). He could only measure “gross weight” and “specific weight” which we would now call mass and specific mass (relative to water, and including the effect of the displaced fluid). Since the local value of g is constant, these are exactly proportional to what we now call weight. It might seem that Earth-bound engineers thought of “weight” as a force of gravity, but the heaven-bound astronomer Kepler thought of it as mass, as we shall see. It is on the basis of the above discussion that Kepler's ideas of mass and inertia can now be judged.

Because the word “weight” properly meant what we now call mass, it is difficult to see how any undefined word or phrase (e.g., mole, “amount of matter” or “quantity of matter”¹⁴) could have meant mass without defining it as “weight.” Kepler¹⁵ actually did this, when discussing the causes of periods of planetary motion in his Epitome where he states: “secunda pondus seu copia materiae transportandae” (“the second [cause] is the weight or the amount of matter (copia materiae)¹⁶ to be transported”) and “pondera vero, seu copia materiae in diversis Planetis” (“weights in truth, or the amount of matter in the different planets”) which means that Kepler defined the amount of matter in the planet as its “weight”—which we now call mass. Clearly, he did not mean here what is now known as weight (as a force). So Kepler did form a concept of mass, but called it “weight” as did everyone at that time.

About 65 years later, in an unfinished manuscript of the Principia, Newton¹⁷ wrote almost the same idea and using the same phrase, “amount of matter,” as Kepler used:

At this time Newton had recognized that weight could mean two different physical quantities, which is the same point at which Kepler arrived in the above definition of weight as copia materiae. Newton had not yet decided how to rename the two components. Ultimately, he called the force of gravity weight and applied the new name mass to what he defined as “quantity of matter.”¹⁸ In Newtonian terminology then, the beam balance measures mass; it cannot and does not measure weight or the force of gravity, mg, because the measurement is independent of the value of g. Kepler's definition of weight (as mass) anticipates Newton's first definition of weight (as mass).

Following Jammer's¹⁹ translation of a passage from Astronomia Nova of 1609, Hecht suggested that Kepler may have used moles to mean mass. However, this was certainly not true in 1620 where Kepler differentiated between mole and pondere when discussing the revolution of planets stating “quanto sunt expeditiores ad motum, densitate, mole, pondere” ²⁰ (inasmuch as they are readier for movement by reason of their density, bulk or weight).

Working from Tycho Brahe's remarkably accurate observations, Kepler discovered three laws of planetary motion, from the last of which Newton derived the inverse square law component of his universal law of gravitation.²¹ Thus, Kepler was probably one of Newton's greatest influences. However, Kepler's laws formed a purely empirical geometric model and did not constitute a physical model.

Kepler then set out to devise a physical model for the behavior of the planets. In the second edition of Mysterium Cosmographicum in which he added notes representing 25 years of experience after the first edition—Kepler introduced a “capacity of resistance” or inertia which is associated with the “weight” of the planets:

Here again, Kepler uses “weight” to mean mass, rather than force, and associates that mass with inertia. His statement is remarkably similar to Newton's definition of quantity of matter (as volume x density²³).

However, Kepler's “inertia” was a force resistant to continuous motion, not to change of motion, and operated as a retarding force on a body undergoing continuous motion caused by some externally applied force, in this case by some sort of rays from a rotating sun. It is essentially based on the belief that it is harder to continuously push (without acceleration) a heavy body than a light one. It therefore follows the Aristotelian idea that any continuing motion must be sustained by the force of a mover, and if the moving force is removed, then the body will come to rest. This model of inertia failed because it followed ancient philosophical teachings which Newton clearly rejected saying “I do not mean Kepler's force of inertia by which bodies tend to rest, but the force of remaining in the same state whether of resting or moving.”²⁴ In any case, some eight years earlier, Galilei²⁵ had already introduced the concept of inertia (but did not name it) in quite opposite terms—as something that would maintain the circular motion of a ship around the earth provided all external impediments were removed. In the present case inertia would maintain the planets on their course around the sun in the impediment-free space.

Kepler was drawn between rational based philosophy and evidence based science. His three laws of planetary motion were brilliantly scientific, and were the first complete evidence for a solar centered system. Newton used them in determining his inverse square law of gravity. Kepler promoted the idea that the planets were material bodies, having all the properties of ordinary bodies. Most importantly, he realized that “weight” really meant what we now call mass, using the term “amount of matter” (copia materiae) in place of the rather vague philosophical term “quantity of matter” (quantitas materiae) which had no scientific meaning. About 65 years later, Newton virtually mirrored Kepler's words, and made “quantity of matter” synonymous with Kepler's “amount of matter” which he then named mass and defined further by his now famous second law. Thus Kepler anticipated Newton's definition of “quantity of matter” and could well have been instrumental in suggesting the idea to Newton, particularly as Newton originally used the same phrase (copia materiae) before naming it “mass.” However, Kepler's model of “inertia” as a capacity to resist motion rather than acceleration was based on Aristotelian thinking and was rejected by Newton.