[libertine]

Tough call. Rayon can be made to look like just about any other fiber. If the cloth is very old, into the 1800s, it is probably silk, as rayon was first manufactured in the late 1890s.
Just as a practical guess, rayon is likely, as it is more durable than silk for something like a table cloth.
Take it to a fabric store. Someone there can probably tell you, or tell you how to find out.