System of polishing external pieces for a timepiece

A system of polishing a concave surface of an external piece for a timepiece, comprising:

• a securing device including a support that carries only the external piece;
• and
• a grinding device including an abrasive mechanism formed by a disc rotatably mounted along a first axis to polish the external piece along a first curvature, the disc being substantially cylindrically shaped and including a first face and a second face with a peripheral wall therebetween, an orientation of the disc defined by a plane perpendicular to the first axis and equally dividing the disc;
• wherein the securing device further includes a moving mechanism of the support so that the support imparts a repeated, oscillating motion around a second axis that is located within the plane, the external piece remaining in contact with the disc during the repeated, oscillating motion;
• wherein a contact surface of the abrasive mechanism formed on the peripheral wall of the disc is curved to polish the piece along a second curvature in addition to the first curvature, and
• wherein the peripheral wall includes the first curvature and the second curvature.