Piano shimmy can usually be attributed to loose legs, and it may be
difficult to tighten them sufficiently with the piano's weight still
on them.  If possible, use a car jack and suitable spacers to take
the weight off of each leg in turn.  If they are held in place with
wood thread screws and will not tighten _very_ firmly, you should plug
the holes with hardwood dowels and re-drill with suitable pilot holes.
If held in place with machine thread bolts, they should tighten
sufficiently.

In either case, if the piano still shimmies when weight is returned,
then it's likely that one or more of the various glued joints of the
leg's components have broken down, in which case the leg(s) will have
to be removed, knocked apart, and reassembled with modern wood glues
such as Titebond, etc.  This is one application where there are better
choices than hide glue.

Right now I have a customer's Weber Duo-Art up on cinder blocks
while all of the legs are rebuilt.  Curiously, it was not shimmying,
possibly because it also has a "stretcher" style base (not my favorite
to work under), but noticeable vertical cracks had developed in the
turned and carved portions.  The joints at the "lintel" (top) of
the legs, a frequent failure point, may still have been okay but the
woodwright is doing the complete job.

John Grant

 [ At http://www.britannica.com/technology/post-and-lintel-system 
 [   In building construction, the post-and-lintel system is a system
 [ in which two upright members, the posts, hold up a third member,
 [ the lintel, laid horizontally across their top surfaces.
 [
 [ At https://en.wikipedia.org/wiki/Capital_%28architecture%29 
 [   In architecture the capital (from the Latin caput, or "head",
 [ Greek kapita) forms the topmost member of a column (or a pilaster).
 [ It mediates between the column and the load thrusting down upon it,
 [ broadening the area of the column's supporting surface.
 [
 [ -- Robbie