[andrewla]

I don't think this is true. Archimedes screw is a similarly shaped helix but contained or partially contained in a tube so that it can move substances by rotary action.

Using a helix as a simple machine is an old concept and probably predates writing; that's about as much of a common ancestor as they share.

[dtgriscom]

Agreed. Archimedes' screw depends on gravity pulling the water to one side of the mechanism; it can't work vertically.