In his first two laws of motion, Newton was merely dealing with the inertial mass of a material body which only had one primary role: to resist any change in its state of rest or its state of uniform motion in a straight line. Newton was also merely dealing with an external force which acted upon that inertial mass (resistance) and accelerated it in one certain direction. Newton’s third law of action and reaction was merely a consequence of his first two laws. All three of these laws were simple and easy to explain and each could be dealt with separately and independently.

On the other hand, the gravitational interactions of two or more material bodies are completely interdependent and each body appears to play several different roles: i.e. of an energy force, of a mass of inertial resistance, of an acceleration and/or a weight. These interdependent and contemporaneous interactions are much more complicated, confusing and much more difficult to describe and explain. Not only that, but (very importantly) such gravitational interactions had to be described and explained by Newton so as to be consistent with the paradoxical results of Galileo’s Leaning Tower of Pisa experiment.

Newton’s attempted descriptions and explanations of these gravitational interactions were both a monumental achievement and series of confusing gravitational concepts, which were often described and explained with ambiguous terminologies. Some of these gravitational concepts were correct, some were partially correct, some were not correct, and some were meaningless. Some of the necessary concepts that should have been included were left out. Other concepts that should have been left out were included. No wonder that there has been so much confusion with Newton’s and Galileo’s concepts of gravity.

Newton’s first five Propositions in Book 3 of the Principia were primarily devoted to describing the force of gravity that causes all bodies to gravitate toward each other, often in orbital motions. In effect, Newton was logically constructing his argument that gravitational attractive force is universal. It was not until Propositions 6 and 7 of Book 3 that Newton attempted to explain the details of how and why such gravitational interactions and accelerations occur.

At the beginning of Proposition 6 of Book 3 of the Principia, Newton stated the following theorem:

“All bodies gravitate toward each of the planets, and at any given distance from the center of any one planet the weight of any body whatever toward that planet is proportional to the quantity of matter which the body contains.”

In this theorem, Newton was attempting to describe the weight of the unequal cannonballs which accelerated toward the Earth in Galileo’s Leaning Tower experiment.

One might ask: What did Newton mean by the word “gravitate”? Newton states that he meant “accelerative gravities” or “accelerations,” and he used all three of the above words to describe the same phenomenon of “mutual gravitational accelerations of bodies” throughout Proposition 6 of Book 3.

What did Newton mean by the word “weight” of a body? Newton states that he meant “heaviness,” “heaviness toward,” “gravitate toward,” “falling toward,” “accelerative gravities toward,” and “weight toward another gravitating body or mass.” And he also used all of these different words and phrases to describe the same phenomenon of “mutual gravitational accelerations” of bodies in Proposition 6 of Book 3.

Actually, the concept of “mutual gravitational accelerations” only occurs when the forces exerted by two opposing inertial masses in space mutually pull on each other, and the concept of weight (the sensation of heaviness) only occurs when a pulled body’s gravitational acceleration is restrained by an atmosphere or is terminated by a solid surface. The fact that both of these concepts are proportional to the inertial mass of the pulling body and are inversely proportional to the inertial mass of the pulled body (as we shall soon discover), only serves to illustrate the validity of Newton’s three laws of motion gravitationally interacting together. But Newton only rarely mentioned any of his three laws of motion in Book 3.

Instead, Newton deferred to Galileo’s Leaning Tower of Pisa experiments in Proposition 6 of Book 3, with the following statement:

“Others have long since observed that the falling of all heavy bodies toward the earth…takes place in equal times, and it is possible to discern that equality of the times, to a very high degree of accuracy, by using pendulums.”

Why was Newton so reluctant to describe the mutual gravitational interactions of gravitating bodies in space by using his three laws of motion? The answer to this question is obvious to the author.

Newton was completely fooled by the paradoxical equal results of Galileo’s Leaning Tower of Pisa experiment, where the unequal masses of two non-identical cannonballs appeared to fall to Earth in apparently equal times. For this reason, Newton decided not use his three laws of motion in conjunction with his universal law of gravitational attraction to describe the gravitational interactions, motions, gravitational accelerations and weight of inertial masses in space, because the application of such laws would contradict the equal results of Galileo’s Leaning Tower of Pisa experiment.

Therefore, Newton was forced to imagine new concepts that would appear to be compatible with Galileo’s paradoxical equal result, and which he could use to rationalize such equal result. To achieve this goal, Newton invented two new concepts of mass for a gravitating body. This endeavor actually resulted in an independent trichotomy of masses.

First, there was the mass of a body which contains and exerts centripetal force in order to attract or induce another body to gravitate. Newton described this category of mass in Proposition 7 of Book 3, and he stated that it was proportional to the quantity of matter (or “inertial mass”) contained in each body. Much later, this energy mass concept (which results in centripetal force) was given a name: “active gravitational mass.”[1]

Secondly, there was the weight or heaviness of a body toward another body which Newton described in Proposition 6 of Book 3 as proportional to quantity of matter or “inertial mass” contained in each heavy body. Newton then empirically determined this proportionality to be equal when he compared the motions of two identical pendulums (with equal weights) to be exactly the same.[2] Much later, this equality of weight and inertial mass of the same body was referred to as the susceptibility of the mass of a body to receive gravitation, and it was given a name: “passive gravitational mass.”[3]

Thirdly, there was our old friend, inertial mass: the resistance of a body to being moved. But, strangely enough, Newton never referred to the inertial resistance of the pulled mass as being inversely proportional to the force exerted by the pulling mass.

The concept of Newton’s three laws of motion interacting together was completely missing in the gravitational context.[4]

There were good reasons why Newton used the ambiguous word “proportional” to refer to the gravitational concepts and gravitational interactions of force, inertial mass (resistance), gravitational accelerations, and weight. All of these concepts and interactions are proportional or inversely proportional with respect to any opposing gravitating inertial masses, but only some of them are equal (or equivalent) as we shall soon discover when we describe Newton’s laws of gravitation the way that they should have been described: as ratios of quantities.[5]