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The Circularity of
Force = Mass x Acceleration

by

July 31, 2020

Introduction

One of the earliest equations to be presented in any elementary physics course is Newton's second law, F = ma, where F is force, m is mass, a is acceleration, and ma means the product of the mass and acceleration.

A student of math and logic might ask a question about this equation that has been asked countless times in the history of physics courses: "How are we to understand the meaning of mass or force if they are defined only in terms of each other?"

This article is about clarifying well precedented but misleading characterizations of empirical observations. It is about correcting the historically understandable but imperfect use of language. A scientific definition needs to name the term being defined, and describe the meaning of that term only by reference to meanings previously established. In physics, the description usually includes how the meaning is to be quantified. Achieving all of these features has been made difficult by descriptions engrained in our language.

Physicist David Goodstein in his lectures in the series The Mechanical Universe Episode 6: Newton's Laws, comments on this circularity, yet unresolved as of then (over 300 years after Newton). See the part starting at the 25 minute point in the video.

As the immediately following sections explain, time and distance, and thereby speed and acceleration, have been quantified without contradiction or circular description.

On the basis of their historical usage and the observable facts, it is shown that formally correct definitions of mass and inertia can be constructed without reference to the term force.

The equations in this article refer to both mass and distance. The letter m has been use to mean both, and has thereby become dependent on the context. It will here be used to mean mass. Distance, being measured in meters, will be referred to with the term meter, or the term meters.

Quantifying Time

A radioactive Caesium 133 atom at rest emits radiation. This radiation corresponds to the transition between two hyperfine levels of the ground state of the atom. The period of the electromagnetic wave emitted when the atom undergoes this hyperfine transition serves as the basis for the quantification of time. When this period is multiplied by 9,192,631,770 the resultant interval of time is the international standard for one second.

To understand how this is expressed in the form of an equation, the historical concepts of frequency and wavelength must be introduced. When undergoing the hyperfine transition, Caesium 133 radiates an electromagnetic wave that crests 9,192,631,770 times per second. This has been called a frequency of 9,192,631,770 cycles per second (in the same way that sound frequencies are characterized). Nowadays it is said to have a frequency of 9,192,631,770 Hz, in honor of Heinrich Rudolf Hertz who was born on February 22, 1857 in Hamburg, Germany and died January 1, 1894 in Bonn, Germany. He showed that Scottish physicist James Clerk Maxwell's theory of electromagnetism was correct, and that light is a form of electromagnetic radiation.

We also speak of the wavelength of electromagnetic radiation. Physically, it is the distance traveled through a vacuum during one cycle of the electromagnetic radiation.

Given all this historical terminology concerning sound and electro-magnetic radiation, the international standard second is this:

9,192,631,770
1 sec     =       ______________
∆ νCs

The term ∆ νCs is the frequency of the radiation emitted by a Caesium 133 atom when undergoing the hyperfine transition. One reads ∆ νCs as delta nu sub Caesium.

The foregoing description of time is forgivably incomplete as a formal definition. It quantifies time, but it does not first define it. This is because there is no person who might read this definition who has any doubt as to what time is; but this kind of omission cannot be made in the case of mass, or probably in the case of any technical terms, other than time and distance.

Quantifying Distance

The international standard meter is the length of the path travelled by light in a vacuum during a time interval of 1/(299,792,458) seconds.

This quantification relies on the constancy of the speed of light. Strange as it may seem, many experiments, beginning with the Michelson–Morley experiment, have shown that the speed of light in a vacuum, as measured by the receiver of that light, is the same, regardless of the speed or direction of travel of the source of that light.

A speed equals a distance traveled divided by the time elapsed. It may seem that we are depending on a definition of distance, but if we arbitrarily choose a distance to be a meter, and arbitrarily choose a duration to be a second, and we use those meanings for distance and time, the receiver of the light will always observe the same speed, regardless of the frame of motion of the source of that light. You can change the length that you are calling a meter and do another thousand calculations of the speed of light. In that event, you will find that each of those thousand calculations give you the same result, whatever it may be. You can change what a second is and calculate a different speed result, but if you measure the speed of light another thousand times with this new meaning of the second, all one thousand results will be the same. So, as long as we define speed as being distance divided by time, the speed of light is constant.

It therefore makes sense that the international standard meter is the length of the path travelled by light in a vacuum during a specific fraction of a second, provided we know what a second is (as we do).

Knowing the duration of a second and the length of a meter, the speed of light, c, has been quantified as 299,792,458 meters per second. This leads to the following quantitative derivation of the meter:

c(sec)                   299,792,458 meters(sec-1)(sec.)
__________    =    ____________________________  = 1 meter
299,792,458                             299,792,458

Speed is defined as the number of meters traveled per second (distance divided by time). Acceleration is defined as the rate of change of speed, measured in meters per second per second, (meters x sec-2).

Light of all frequencies (we see them as colors), travel at the same speed; but so do all of the other frequencies of electromagnetic radiation. Radio waves, microwaves, and infrared waves, have frequencies lower than those of visible light. Ultraviolet waves, X rays, and gamma rays have frequencies higher than those of visible light. All of these radiations travel at 299,792,458 meters per second in a vacuum.

Every person who learns the language of their parents has had experience with the words more and less, and then distance. By witnessing what is called more distance and less distance, the concept of distance is realized.

The foregoing description of distance quantifies distance, but it does not first define it.

In physics, quantifying whatever is named is a high priority, but with rare exceptions, a formal definition needs to define a term. This feature of the formal definition can be absent only if one is absolutely certain that the reader knows what the meaning is (as we are of time and distance).

Quantifying Mass

Max Karl Ernst Ludwig Planck was born April 23, 1858 and died October 4, 1947. He defined Planck's constant. Although he used different but equivalent units, Plank's constant has this exact value:

h = (6.62607015x10-34)kg(meters2)(sec-1)

The utility of this constant and the insights and successful experiments that it has inspired are so impressive that it is now taken as an axiom of physics, and has created agreement on the quantification of the international standard kilogram:

h(meters-2)(sec)              (6.62607015x10-34)kg(meters2)(meters-2)(sec-1)(sec)
_______________    =     ___________________________________________    =   1 kg
6.62607015x10-34                                   6.62607015x10-34

Observing that some particular thing has a specific quantity of mass does not tell us what mass is. A definition of mass would tell us what it is, but this is a quantification of mass.

There is a more serious problem. This description of mass presumes that force is previously defined. The unit of force is the Newton. Newtons are measured in kg x meters x sec-2. This can be factored out from the equation above.

After mass is defined and quantified without reference to force, this quantification can be used for its greater accessibility and precision over the previous quantification, but it cannot be a way of defining mass in the first place.

In 1775, a standard of mass was defined in order to compare it to objects of unknown mass. The standard then was as a quantity of water. Later, it was an object made of platinum; then (and until 2019), it was a cylinder made of platinum and iridium. See the details here. This kind of quantification does not require introducing the concept of force. After mass is provided with a formal definition that does not depend on the concept for force, the Planck's constant method can be used to quantify mass.

The history of accelerations of common objects does not show us a change from one speed to a measurably different speed without these speeds happening at measurably different times. No change of speed of common objects happens instantaneously. The creation of light or other electromagnetic radiations results in an instantaneous change to the speed of light upon the creation of that light, whereas common objects (objects comprised of atoms), always accelerate gradually (whether the succession of these gradations is rapid or slow). The acceleration of common objects is always a process that goes through time. This applies to gravitational accelerations as well.

Rates of Acceleration

A magnet may accelerate toward or away from some apparatus. The structure and cargo of a rocket may accelerate upward. Charged particles may accelerate in a cyclotron. A projectile may accelerate along the inside of a tube (with expanding gases following it). You may throw a ball, or pedal a bicycle from a stop to a non-zero speed. Any person can accelerate a base ball to a higher speed than a steel ball that is four inches in diameter. Cartridges being equal, an aluminum projectile will accelerate faster than a lead projectile.

Whenever this is observed, we are confronted with yet another characteristic of objects that is consistently exhibited. Their acceleration is limited. Their acceleration is limited as surely as it is compelled. Under such circumstances, some objects limit their rate of acceleration more than others do.

Both the acceleration and its limitation are characteristics of the motion of these objects.

We must attribute these characteristics of their motion to the objects themselves. This is confirmed by the fact that different objects generally accelerate at different rates of acceleration. Accelerations are not the same for all objects.

Whatever property of objects exists that accounts for limiting acceleration, we can give it a name. I would name it inertia. There is no need to include the concept of force in the definition of inertia.

The property of gravitation is proportional to the magnitude of the masses of the objects being accelerated toward each other. It is a separate property of objects.

Definitions

There are three words in physics that are at the core of the current dilemma. Acceleration is not one of them. We know what a second is, and we know what a meter is. These three words are mass, inertia, and force.

A formal definition names the term to be defined and describes what it is. Also, it is the preference of physicists that it specifies how that meaning is to be quantified. It is not adequate to spell out how a term is to be quantified without assuring that the reader knows what it is.

When a formal definition describes what a term is, it must not mention any term not previously known. (That includes the term being defined.) Terms in the description could be a part of the language agreed by everyone who speaks the language, like the or going; or they could be terms defined formally, that is, constructed from meanings previously established.

Exact wording aside, definitions of inertia universally express the same idea. They all have the same defect. They mention the word force in the absence of a formally correct definition of force. The definition of force mentions the word mass in the absence of a formally correct definition of mass. Definitions of mass almost invariably mention the word force, while force is not previously defined.

Here are some well-publicized definitions of mass:

"Mass, in physics, quantitative measure of inertia, a fundamental property of all matter. It is, in effect, the resistance that a body of matter offers to a change in its speed or position upon the application of a force. The greater the mass of a body, the smaller the change produced by an applied force." Here is the source.

"Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. Here is the source.

"The mass of a body is its inertia or resistance to change of motion." Here is the source.

"Mass is defined as a quantitative measure of an object’s resistance to acceleration." Here is the source.

There are two definitions that do not refer to force, but one of them refers to inertia. Here is how inertia is currently defined:

To quote Merriam-Webster, "Inertia is a property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force."

This definition of inertia depends on the prior establishment of the meaning of force. Since we do not yet have an independent definition of force, we are, once again, defining two important ideas with only one definition.

In a discussion of inertia, another author has describe inertial mass this way: "The inertial mass of an object is its resistance to a force." See The Equivalence Principle: A Question of Mass

It may well be that an object's mass is to be associated with a limitation on its acceleration, but once again, force is being treated as a known quantity.

Only definitions of mass that do not refer to force will eliminate the circularity that has troubled so many.

From the considerations explained above, it is clear that inertia is the property of objects that limits their acceleration.

Mass is identical to inertia. Mass is the property of objects that limits their acceleration.

We simply have two names for the same property of objects.

Again, the gravitational property of objects is proportional to the magnitude of the masses of the objects being accelerated toward each other. It is a separate property of objects.

The first law states that every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it. This is not a definition of inertia. It is not a definition of mass or force. It is a statement that Newton asserted as true. He did not offer formal definitions, much less did he offer them in the order of their mutual dependence.

Early in his Mathematical Principles of Natural Philosophy, Newton described mass:

"The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter."

This is a description of matter and not a description of the property of matter that is known as mass. We cannot know his concept of this property. Others subsequently made their presumptions.

The foregoing definition of mass can be quantified without reference to force. As in the past (when we had no formal definition of what mass is) we can select a standard to which other objects can be compared. With the balance scale (described below, and which gives us the same comparison on any planet) we can compare any object to this standard object which we know as the kilogram. Then, we must see if all objects measured as a fraction or multiple of the kilogram limit acceleration in proportion to their balance scale measurement. This is done by utilizing whatever other factors, circumstances, and changes are also associated with the occurrence of acceleration, and selecting those found to be repeatable.

If in all such circumstances the proportions measured by the balance scale indicate the degree to which objects limit acceleration, then mass is that which is measured by the balance scale, using the kilogram as the standard mass; and the mass of an object is measured in kilograms.

The mass of objects can be compared to that of a standard by means other than the balance scale. Electromagnetic apparatus creating repeatable accelerations in the standard object could demonstrate different accelerations in objects of unknown mass. Any apparatus creating such repeatable accelerations could be used to compare masses.

Force has always been presumed to be the cause of all accelerations of objects, gasses, and particles (whether charged or not). This cannot be proved. We can say that force is associated with the acceleration of common objects and is the apparent cause (in the statistical sense) of the acceleration of common objects.

At this point we can quantify the force seen as responsible for the acceleration of common objects as being equal to the mass of those objects times their acceleration. In equation form, we have the following:

F = ma.

Since mass and acceleration were defined and quantified before the formation of a formal definition of force, the unprovable or magical understandings of force is of no consequence. This is because force is now a derivative quantity. It is a quantity first and foremost, and it is derived from mass and acceleration as their product. We may believe that force is the cause of acceleration, but there is no need to prove it.

One should be clear on the meaning of cause. There are three requirements that must be met to establish a cause-effect relationship in statistics between event 1 and event 2: First, event 1 must occur before the event 2. Second, event 2 must occur whenever event 1 occurs. Third, another explanation for the association of event 1 with event 2 does not exist. Only then is event 1 said to be the statistical cause of event two.

The third requirement is a negative which can never be proved.

Informally, alarm clocks are usually the cause of arousal. Physics is more serious than that. Physics can shape expectations by demonstrating what is associated with what. There is no need to name or describe or define causes. It's a hangover from the days of magical thinking.

Final Quantification of Mass

Now that mass and force are well defined, we can used a method of quantifying mass that is more accessible and more accurate than the method which uses a physical object as a standard of mass. The Planck's constant method, adopted as the international standard of mass, suffered from the failing that it presumed the existence of force, in the absence of a formal definition of force.

As with mass, force now has a correct formal definition; therefore, the Planck's constant method (which assumes that force is previously defined) is a perfectly legitimate method of quantifying mass:

h(meters-2)(sec)              (6.62607015x10-34)kg(meters2)(meters-2)(sec-1)(sec)
_______________    =     ___________________________________________    =   1 kg
6.62607015x10-34                                   6.62607015x10-34

Interpreting Isaac Newton

Could this analysis be too simple? This section examines Isaac Newton's laws of motion and their possible consistency with the above proposed definitions of mass and force.

As calculated by our modern calendar, Sir Isaac Newton was born on January 4, 1643 and died on March 31, 1727. Among his very many other accomplishments, he laid down the foundations of classical mechanics in his 1687 book Mathematical Principles of Natural Philosophy. Here are Newton's laws of motion:

The first law states that every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it.

The second law states that, for an object of constant mass, the force on that object, F, is equal to its mass, m, multiplied by its acceleration, a. As an equation, this is written as F = ma.

The third law states that for every force, there is an equal force in the opposite direction.

Newton also defined a law of universal gravitation:

G(m1)(m2)
F    =      ___________
r2

F is the force of attraction between two objects having constant masses, m1 and m2; r is the distance between these objects; and G is Newton's gravitational constant.

Newton's 1687 book contains definitions, one of which I feel is very germane:

DEFINITION III. The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertiæ, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

The first sentence is worth repeating: "The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line."

The word force in this sentence is not the force which is seen as the cause of acceleration. It is a property. The "innate force of matter" is the power (ability) to resist a change in the state of motion of that that matter. Newton declares vis insita to be a synonym of this property, which has been widely termed inertia. Whether this is a definition of inertia or mass, it is a definition which does not rely on force, having been previously defined. It is independent of the concept of force as the cause of acceleration. If this was Newton's understanding of inertial mass, then his second law is without contradiction or circularity.

Some experiments known to Newton had been performed by Galileo Galilei, who died almost a year before Newton was born. According to our modern calendar, Galileo Galilei was born February 26, 1564 and died January 8, 1642. His experiments importantly showed that objects fall to Earth at the same rate of acceleration, provided that air resistance is neglected. (Air resistance is obvious in the case of objects like paper and feathers).

The second law (F = ma) is a statement about any force, regardless of the source. The concept of force is found in every law of classical mechanics laid down by Isaac Newton.

In Newton's law of gravitation, there is no indication that Newton regarded gravitational mass as a different form of mass from the mass referred to in the second law (F = ma). Any more recent proof that gravitational mass and the mass of the second law are the same is superfluous. There is no doubt that the meaning of mass being referred to by Newton in the quantification of gravitational force is the same meaning referred to in the second law.

A scientific law is not a scientific definition. The circularity of F = ma is not due to Newton's laws.

The term inertia has become highly associated with the current understanding of mass. Many have interpreted Newton's first law as an improvement on earlier ideas about inertia. This gave rise to definitions of inertia that use the word force, incorrectly presuming that force had been previously and independently defined.

Some noticed the use of the word force in Newton's first law of motion and took it to mean that which changes the state of motion of a body. Force has long been presumed to be the cause of motion. It was taken to be the cause of the acceleration of all substances and things.

Instantly perceiving that there is a force at play wherever there is evidence of an acceleration is the result of an over-learned association. This is the specific and unfortunately widespread conditioning that interferes with accepting a definition of mass that does not refer to force (presuming that force is the only possible cause and that one must mention it).

For centuries, the concept of mechanical force has been discussed by philosophers and scientists. They include Archimedes, Aristotle, Galileo Galilei, Isaac Newton, and others. Many centuries earlier, words for mechanical force were in the lexicon of every human language.

As evidence of this, the word force, in the sense of mechanical force, is found in Book V of The Odyssey, by Homer. The English text, written down by S. H. Butcher, M.A., includes the following phrase: " . . . the force of the wet winds blew . . . " The Odyssey is widely believed to date back to the Eighth Century B.C.

From ancient times, language expressing the idea of mechanical force has been given meaning through several gradations of muscle strain. This force is not always experienced to the same extent or very accurately. We have few words describing how much force, yet we have the illusion of knowing what force is.

The widespread belief that force is the only possible cause of acceleration is not something that can ever be proved. Believing this is not necessarily harmful, but if it prompts one to jump at the chance to explain causation - not in a report, or an explanation, but in a definition the logical discipline of a formal definition is compromised.

This happens when we say that the definition of inertia is the property of common objects that prevents their acceleration unless they are accelerated by a force.

You could just a well say "unless moved by the divine will of the goddess Anu".  At least then, everyone would say "You can't prove that."

At one time, it was presumed that all things that accelerate do so gradually and under the influence of a force. This now seems to be true of objects comprised of at least one atom, but not light, which attains the highest known speed instantaneously upon its creation. Light passing close to a star bends its path because of its relativistic limitation of acceleration (a change in direction is an acceleration). Radiations acquire mass as a result of their high-speed motion. All objects acquire additional mass as a result of attaining speeds that are a noticeable fraction of the speed of light.

Educators may hasten to explain why, but formal definitions are about what.

By Newton's time, scientists did not confuse mass with weight; but historically, mass got its start in language as weight. The two terms are still often conflated.

The concept of weight is very much associated with the qualitative experience of force. It is sensed by the body, but it's hard to quantify verbally. The force felt when lifting something may not have been accurately quantified; nonetheless, it has been referred to for many centuries as weight. Various devices have been invented to make weight more objective.

At around 5000 B.C., Egyptians invented the balance beam for weighing grains. See one here

The steelyard balance was independently invented in several countries in ancient times. According to Thomas G. Chondros of the University of Patras in Greece, the steelyard balance was used by Greek craftsmen of the 5th and 4th centuries B.C. See the steelyard balance here

The daily experience of picking up common objects or holding them to prevent them from falling toward the ground or the floor is due to their tendency to accelerate toward the Earth. Consider two objects that I shall call Ob1 and Ob2. We will presume that these objects are small enough to be manually lifted and held. Such objects have been compared by means of instruments such as the balance scale illustrated below: The black rectangles represent Ob1 and Ob2. They are each suspended by a green string. There is a black circle at the top of the central green pillar that represents the end view of a cylinder that is attached to the horizontal beam by means of a flexible strap (not shown). This gives increased leverage to the side of the lever that is raised, should there be a minor imbalance. When the horizontal beam is exactly horizontal, the balance is found.

There is a measurable ratio of the distances from the top of the cylinder to the green strings attached to the horizontal beam. This ratio can be used in two ways. Of course, the first use in history of any balance device was to discern the weight of an object. This was done by declaring a special object as a standard of weight, and then comparing that standard to the object of yet unknown weight.

(In the balance scale pictured above, the tendency for Ob1 and Ob2 to accelerated toward each other is many orders of magnitude less than their tendency to accelerate toward the Earth. This would be represented by a thoroughly undetectable difference between the direction of the string's length and the vertical.)

Remarkably, the balance scale would show us the same comparison on the moon, despite the fact that these objects would accelerate toward the moon at a much lower rate of acceleration when near the moon's surface than they would accelerate toward the Earth when near the Earth's surface. This same ratio could be calculated from the measurements of the steelyard balance as well. The Egyptian balance beam could only show an equal balance. The user of the balance beam would add small quantities to one tray until it balanced the weight of the other tray.

The balance scale pictured above directly compares masses and gives the same ratio on any planet. With the introduction of a weight standard, either the balance scale or the steelyard balance can be used to measure weight. The Egyptian balance only detects when two separate amounts of material have the same weight.

The ratio of two masses (more exactly, the ratio of the mass of two objects) could be defined as the ratio measured by the balance scale, but this does not quantify the mass of any particular object, unless we declare a particular object to be the standard of mass against which objects of unknown mass can be compared.

Notice also that when two objects of different mass accelerate toward each other, the object having the greater mass accelerates less than the object having the lesser mass. Further, each acceleration is consistent with the second law (F = ma). For objects a and b, we have the following:

Fa = ma x aa

Fb = mb x ab

Of course, their accelerations are in the opposite direction. The absolute values of Fa and Fb are equal.

Let us take a closer look at the gravitational force.

In the equation F = ma, acceleration has the units of meters per seconds squared.

a = meters/sec2

The Newton, N, is the unit of force. A Newton is the amount of force needed to accelerate one kilogram of mass at the rate of one meter per second squared.

1 Newton = (1 kg)(1meter)(sec-2).

G is the gravitational constant:

G = 6.67430 x 10-11(meters3)(kg-1)(sec-2)

Again, here is Isaac Newton's force of attraction of two objects having masses m1 and m2:

G(m1)(m2)
F    =      ___________
r2

Showing the units of measure for the masses, m1 and m2, and the distance, r, gives us this equation:

G(m1)kg(m2)kg
F    =      ______________
(r meters)2

Showing the units of measure for the gravitational constant, G, we have the following:

6.67430 meters3(m1)kg(m2)kg                 6.67430 meters3(m1 )(m2) kg2
F    =      __________________________    =      __________________________
(1011)(r meters)2(kg)sec2                           (1011)r 2(meters2)kg(sec2)

6.67430 meters (m1)(m2) kg
F    =      ________________________
1011r2sec2

We see that force is indeed in units of (kg)(meters)(sec-2).

Since a = F/m, the acceleration of m1is this:

6.67430 meters(m2)
am1    =      __________________
1011r2sec2

It also means that the acceleration of m2 in the opposite direction is this:

6.67430 meters(m1)
am2    =      __________________
1011r2sec2

This demonstrates that the quantification of force given in the second law of motion is consistent with quantification of the gravitational force, and therefore the mass referred to has the same meaning in both cases.

It is perhaps interesting that the acceleration of either of these masses depends only on the mass of the other.

Consider the acceleration of either (m1) or (m2) if attracted only toward a third object (m3) at different times:

6.67430 meters(m3)
am1    =      __________________
1011r2sec2

6.67430 meters(m3)
am2    =      __________________
1011r2sec2

They would, in turn, accelerate at the same rate. The acceleration of either m1 or m2 depends only on the mass m3 (not their own, though they may be different from each other).

We can see that the ratio of the accelerations of objects gravitationally attracted toward each other is a measure of the ratio of their masses. While this method is inconvenient, it can be used to specify the mass of objects once a standard mass is declared.

These accelerations were calculated from Newton's gravitational formula describing the force of attraction, but these accelerations can be verified experimentally without reference to the concept of force. We know what a second is and what a meter is. Even with less accurate standards of measurement for distance and time duration, these observations could be made without presuming that the accelerations are prompted by a force. This is a construct that could have been postponed until a formal definition of force could be announced, but Newton was then constructing laws.

If Isaac Newton described the mutual attraction of objects without referring to force, we could have had the following law of motion:

Each of two objects with masses m1 and m2, with a distance, r, between their centers, accelerate towards each other at a rate that is proportional to the mass of the other object and inversely proportional to the square of the distance, r, between them.

A constant, CN is used to form the quantitative description of these accelerations:

CN = 6.67430 x 10-11(meters)(sec-2)

Thus, when two objects of constant mass, far from other masses, accelerate toward each other, the accelerations observed are these:

(CN)(m2)
am1    =      __________
r2sec2

6.67430 meters(m2)
am1    =      __________________
1011r2sec2

(CN)(m1)
am2    =      __________
r2sec2

6.67430 meters(m1)
am2    =      __________________
1011r2sec2

It should also be noted that these quantifications are very accurate only if the speeds involved are a minuscule fraction of the speed of light.

Conclusions

If there had been no attempt to use the word force in the definition of inertia or the definition of mass; and if defining what mass is before quantifying it had been a priority, the hitherto circularity of F = ma would never have existed.

Taking mass to be the property of an object that limits its acceleration, every law of motion laid down by Isaac Newton is entirely without contradiction or circular description.

One might argue that the terms and concepts of physics are not what they were in Newton's time, and that these issues are obsolete. They are not. A scientific definition is a formal definition. The terms of the future are not likely to be as intuitive as time and distance. If each definition is not carefully constructed to follow the rules dictated by logic, then systems of definitions (even if only comprised of three) can retard progress.