Abstract: A lens (L) or the like is placed in a lens holder (16) with a concave end (CC) in contact with a resilient, semi-spherical polishing head member (98) that may or may not be covered by a polishing cloth (PC). The lens holder (16) and lens (L) are rotated in one direction. The resilient, semi-spherical polish carrying member (98) is rotated in the opposite direction. The lens holder (16) and lens (L) are swung sideways to move the surface to be polished along an arcuate path. The concave surface to be polished is in contact with the resilient, semi-spherical member (98). A surface to be polished presses against the resilient, semi-spherical surface (98), causing it to compress. This stores energy in the member (98) that urges the member (98), and any polishing compound on it, into tight contact with the surface to be polished.