A swivelling body (22) which can swivel on a machine frame (20) defines a lifting axis (D) transversal to its swivelling axis (A) along which a lifting carriage (24) can slide past the swivelling body (22). A tool carriage (26) can be adjusted along a spindle axis (E) on the lifting carriage (24) transversely to the lifting axis (D). A tool spindle (28) which can be fitted with a tool (30) is rotatably mounted about the spindle axis (E) on the tool carriage (26). A workpiece carriage (42) which defines with a workpiece holder (44) a middle plane (G) normal to the swivelling axis (A) is transversely guided on the machine frame (20). A feed finger (58) can be intermittently laid on a cutting tooth (12) of a workpiece (10) to feed the tooth forward. A common sensing body (7) can be moved from a rest position to a measurement position next to point (H), at which the swivelling axis (A) crosses the middle plane (G), in order to determine measurement values which represent the distance between the cutting teeth (12) and the workpiece holder (44), as well as the state of wear of the tool (30) or feed finger (58). The thus determined measurement values are converted into signals for controlling the workpiece carriage (42) and the tool carriage (26) and/or the feeding device (50).