Don Weisman writes:
This photo is only to show the orientation of what
is being talked about:

The wood on Snake Leg tables is normally oriented to run from
where it joins the center column to the leg tip. Good wood
selection doesn't actually happen in real life unless it is a
Craftsman made Table as it takes a lot of time and wood to
locate the correct grain in a wood inventory. This high quality
table was no exception. It had been repaired twice before the
customer handed it to us. Their repairs failed because they didn't
know what to do and what they did do.was poorly done.
The table legs were breaking partially because of improper
gluing of the two dowels in each leg. Also, they were breaking in
the same place each time-at the short grain at the column. You
see, when the leg separated from the lowest to the floor dowel
and away from the column, it was strained and it snapped the leg
at the upper dowel which was still in the dowel hole. There were
other damages also.

THE FIX
We inserted (the thin stripes shown in the photo) two (2) cross
grain pieces of wood, and bore oversized holes for the dowels
we epoxied in. Because the shape is difficult to clamp, we hand
held each leg for the epoxy to firm up using Devcons 5 minute
epoxy. Missing splintered off wood was filled with Tootsie.
Because the previous repair people left several broken off
screws deep into the wood which we could not see, we
massacred a very fine quality drill bit which I will charge it's
sharpening to the customer as part of the job.
Don Weisman, TX