[mad aetherist (OP)]

In spiral galaxies the orbital speeds of stars do not appear to follow the expected Newtonian 1/RR relationship, they appear to follow a 1/R relationship.  As there is no good theory for a 1/R relationship the preferred answer is that every galaxy contains hidden mass made up of some sort of Dark Matter.

I looked at the equation for centrifugal force which is   F=Mvv/R. 
And the equation for gravity force is    F=GMm/RR.
This means that the orbital speed    v=(GM/R)^0.5.

But if the equation for gravity force was      F=GMm/R    then   v=(GM)^0.5.
This means that the graph for orbital speed versus radius should be a horizontal line (notice that R is missing)(R is not relevant).   Cosmologists draw prediction curves for orbital speed versus radius in a spiral galaxy based on   1/R  (merely to illustrate)(they don’t believe in  1/R), but these  1/R  prediction curves are not horizontal straight lines, they are a curve dipping slightly below horizontal.  I guess that this is because a spiral galaxy isn’t like a thin disc, it has a bulge.  The bulge introduces some off-plane components of the gravity force & the components perpendicular to the plane cancel leaving just the Cosine.

The plots for the actual measured speeds for any particular spiral galaxy describe a curve dipping down below horizontal a little, following almost exactly their predicted  1/R  curves.
 
On the other hand the predicted curves for  1/RR  drop quickly towards the xx axis & are hopeless.  There certainly is a problem with Newton.

Everyone agrees that the gravity force due to an infinitely long cylinder varies as per  1/R.   A thin slice of that cylinder can represent a spiral galaxy.  If u take away the remainder of the cylinder then praps the thin slice retains that there  1/R  relationship.  Or does it revert immediately to  1/RR?
 
I am mulling this  1/R  stuff.   Suggestions welcome.