Part III.   Problems with Galileo’s and Newton’s Concepts of Gravity
P.   What Does Universal Gravitational Attraction Really Mean?

What did Newton mean by the word “gravity”? In the Scholium to Proposition 5 of Book 3, Newton stated:

“Hitherto we have called ‘centripetal’ that force by which celestial bodies are kept in their orbits. It is now established that this force is gravity, and therefore we shall call it gravity from now on.”[1]

At the beginning of Proposition 7 of Book 3, Newton then stated that:

“Gravity [centripetal force] exists in all bodies universally and is proportional to the quantity of matter [inertial mass] in each [body]…inversely as the square of the distance…And it follows…that the [sun’s] gravity [centripetal force] toward all of the planets is proportional to the matter [inertial mass in each].”[2]

It also follows from the above statements in Proposition 7 that the centripetal force exerted by and between any two opposing masses depends upon the ratio of the two opposing masses. Please refer to Figure 16 to understand the ratios of the different masses in the Solar System. Briefly, the mass ratio between the Earth and the Moon is approximately: Earth = 82 and Moon = 1.

Proposition 7 of Book 3 has been universally interpreted to mean that: “Each body attracts the other [opposing body] with a force of equal magnitude…even if their masses are quite different.”[3]

But how can this be? How can the centripetal force originating or emanating from each pulling body be proportional to both the mass of the pulling body (i.e. Earth) and to the very different mass of the opposing pulled body (i.e. the Moon), and vice-versa?[4]

The reason is because with gravitational force, we are dealing with two separate concepts of magnitude of force: the intensity or strength of the force emanating from the pulling body in all possible directions, and the quantity of such force received by and acting upon each opposing pulled body. In Proposition 7 of Book 3, Newton inferred that the intensity of the centripetal force emanating from a pulling body in all directions is proportional to its own inertial mass, and that the quantity of such centripetal force received by and acting upon a pulled body is proportional to its own inertial mass. Please read the above quote from Proposition 7 again with this interpretation in mind.

Thus, each opposing body pulls the mass of the other body (its inertial resistance) with a centripetal force, the intensity of which is proportional to the mass of each pulling body. The quantity of such intensity of centripetal force which is received by and acting upon the pulled body (its inertial resistance) is proportional to the mass of each pulled body (Figure 17.1).

Implicit in Newton’s Proposition 7 of Book 3 concerning gravitational attraction (and in Newton’s discussion of the “center of mass” of gravitating bodies contained in Book 3, Propositions 12 and 13), is the fact that the smaller a pulled gravitating mass is, the proportionally less will be the quantity of intensity of centripetal force which it will receive from a pulling mass; and the larger a pulled gravitating mass is, the proportionally greater will be the quantity of intensity of centripetal force which it will receive from a pulling mass…all diminished of course by the inverse square of the distance of such masses apart (Figure 16).

Empirically, the opposing smaller mass (i.e. the Moon) receives a proportionally smaller quantity of more intense force emanating from the larger mass (i.e. the Earth), which is equivalent to the greater quantity of less intense force emanating from such smaller mass (the Moon) and received by such proportionally larger mass (the Earth).[5] Thus, the quantity of different intensities of force of attraction reciprocally received by and acting upon any two opposing gravitating bodies are of equivalent or equal magnitude. Cohen describes this phenomenon as follows:

“…between any two bodies…there is a force of attraction that is mutual, and each body attracts the other with a force of identical magnitude…”[6]

We shall call this mutual and reciprocal phenomenon: the “Equivalence of Gravitational Forces Received.”[7] However, as exemplified by Figure 17.2 and Figure 21, this equivalent magnitude of gravitational forced received is always proportional to the mass of the smaller gravitating body.

Also implicit in Newton’s Propositions 7, 12 and 13 of Book 3, is the fact that the relative intensity of the forces emitted, and the equivalence of the gravitational forces received by each body, is dependent upon the mass ratio of the opposing bodies, diminished by the inverse square of the distance between them. All of these concepts concerning the gravitational forces of attraction received are completely relative and reciprocal concepts.