Part V.   Center of Mass and Relative Gravitational Acceleration
DD.   The Generalization of Newton’s Four Fundamental Axioms

With the observed gravitational interactions of binary stars, of the Earth-Moon gravitation system, and of the Sun-Earth gravitational system as our models and guides, we can now modify, generalize, and paraphrase Newton’s four fundamental axioms (which we found at the beginning of Volume 3 of the Principia), as follows:

1. No more causes of the above gravitational interactions should be necessary than are both true and sufficient to explain such phenomena;

2. Causes assigned to gravitational interactions of the same kind must be the same;

3. Those gravitational qualities and interactions of bodies that belong to all bodies on which experiments can be made, should be accepted as qualities and phenomena of all bodies universally; and

4. Laws of nature inferred from such phenomena by inductive reasoning should be considered to be true notwithstanding any contrary hypothesis, until the discovery of other phenomena make such laws either more exact or liable to exception.

All observations and measurements of our gravitating models guide us to the conclusion that the relative gravitational forces and inertial resistances applied by and between any two opposing unequal masses in space is always a ratio of their masses (m), which ratio of masses is inversely equivalent to the ratio of their gravitational accelerations (a), but the relative gravitational forces applied are always diminished by the inverse square of the distance between their mass centers

Therefore, the equivalence that always applies to such two-body gravitational interactions, is:

where m1 is the force and inertial resistance of the larger mass, a2 is the acceleration of the smaller mass, m2 and a1 are always 1, but the relative forces applied are always diminished by the inverse square of the distance between their mass center

The expression

is the inverse square of the distance between their centers.