Newton Principia
Section XII - Proposition LXX - Theorem XXX

have long been aware of a conflict between any repulsion generated from outside the galaxy and Newton's proof that a test mass within a hollow sphere will feel no push in any direction because of cancellation of forces from opposite sides. An inverse square law repulsion would be no different than an attraction and both will cancel out within the galaxy.

I took the time to examine his proof in detail and found it to be perfectly sound and elegantly simple as well (see bottom of page for jpg of that theorem).

Briefly, lines AB and CD are drawn through a test mass (P) within a hollow sphere. Since the triangles ACP and BDP are proportional, the arcs AC and BD are also proportional. Therefore, if we construct a faceted hollow sphere (something like the mirrored ball at a disco), the facets on opposite sides of the sphere will be proportionate.

Then, if the test mass is, say, 4 times more distant from one facet than the other, the more distant facet will be 16 times as large (16 times more massive).

Therefore, by the inverse square law of gravitational attraction, 1 x 1 = 16 x 1/16 and both facets attract the test mass equally. Then making the facets ever smaller, we converge upon a spherical surface. This is a proof by geometrical integral calculus.

I tried "monkeying" around with the "true proportional arc" [Z] to find some conflict but it turns out that any arc drawn in the expected place is equally attracted to the test mass ...
because ... though one is larger than another, it is at the same time farther away ... exactly cancelling the force due to greater mass.

In response to the forgoing ...

Any supposed repulsion by extra-galactic empty space must not vary inversely as the square of the distance from the galaxy. Rather, I must conjecture perhaps a straight inverse distance relationship. This is not unprecedented since the amplitude of electromagnetic waves varies in this manner while the energy associated with those waves varies by the more standard inverse square law.
Or, possibly, no distance relationship (see ntx16a1.htm)

Since I have already given that the gravitational force must vary as 1/r from a single body (and thus ... m1/r   x   m2/r = {m1x m2}/r2 ... between two bodies ... according to my reason) ... we must suppose that the empty space influences (repels) matter as 1/r alone, being that it is, of itself, bereft of matter ... and is therefore ... incapable of participating in a manner similar to that of two mutually gravitating bodies. And further that the interaction of contention is between unlike members and therefore has, as it were, a single hand in the pot rather than two the same.

In this manner:

Let A and B be two galaxies some distance apart and let C be a space between them. Then the attraction AB varies inversely as 1/r2 ... and the repulsions AC and BC vary as 1/r because C contains no mass and therefore if we consider action-reaction, the interaction AC alone would cause C to fly off in the opposite direction at light velocity while A remained stationary.

AB and C

The inclusion of the repulsion BC from the other side "backs up" C thus preventing it from "getting away". So that, in effect A and B attract each other directly and also at the same time repel one another indirectly through the intermediary C.

But ... C obviates the inverse square rule because its insertion between A and B is that of a stick which transmits a pushing force between two people. That is, the length of the stick is irrelevant ... the force is undiminished by separation.

Figures pushing
These two "pushing" examples are equal with respect to force
even though the separation is different.

In this way, the repulsion of matter by empty space is unlike an inversion of the attraction of two gravitating bodies ... it does not vary as 1/r2. And the symmetric distribution of empty space round about matter is not mutually cancelled by opposed parts as prescribed by Newton.

I therefore propose that the repulsion AC varies as 1/r where r is the distance to the center of the "void space".

From : Newton's Principia

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