the leaves or whatever they are called, were elliptical, about 3/4 of an
inch long, 3/8 of an inch wide, and around 1/8 of an inch thick. no spines or
thorns, with smooth edges. overall height was about 4 inches. anyway, the
leaves would fall to the ground in the pot, but never seem to die
one of those still green leaves. some small red hair-like fibers are
sprouting from this one,

From your description, this is not a cactus at all, but some sort of succulent.
We will need more of a description once it recovers. Let the leaf dry for a day
or so, then plant it with the end that was attached (you will see a scar) in
clean sandy soil (cactus soil OK, maybe with more sand added). Keep it just
barely, barely moist in a bright place out of the sun. Under a fluorescent
light is perfect. It should grow some roots and eventually a new plant, but be
patient. Once it starts to grow roots, you can feed it occasionally with very
dilute plant food.


Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)