In short, round die refers to the curvature of the die face. Early United States
coins typically had flat fields as a consequence of the primitive method of die
preparation. Starting with the Christian Gobrecht designs of (1836-40), USA coinage
began to reveal a slight concavity to the fields, though this isn't always apparent on
casual inspection. Concave fields were the result of convex die faces, and this slight
curvature had to be applied in a step that was separate from the actual sinking of the die.
A working die that was ready for polishing was set into a jig with its face upward. The
face of the die was then brought into contact with a polishing disc, or plate, that had a
very shallow concavity to it. When spun against the face of the die, the disc imparted the
same curvature profile, but the result was convex.