A system for cutting the cornea of an eye, includes a moveable member with a cutting blade at one end, a pivot element and a cutting guide restraint. A mechanism for oscillating the moveable member around the pivot element and a cutting guide are configured to engage the cutting guide restraint on the moveable member and thereby limit the degree of angular movement of the cutting blade as the moveable member oscillates about the pivot element. A positioning system is configured to advance the moveable member with respect to the cutting guide such that the shape of the cutting guide determines the shape of a cut made by the cutting blade. A suction ring for stabilizing the cornea and an applanating plate for flattening the cornea usually complete the system.