A lathe exhibits a vertically positioned, motor-driven work spindle on whose lower end are positioned workpiece clamps, an initial compound slide system (19) by means of which the work spindle can move vertically in the Z1 direction and horizontally in the X1 direction, at least one initial stationary tool holder (22), and at least one second tool holder (46) which can execute an advancing action in at least one direction during the machining process. The advancing movement of the second tool holder (46) is independent of the control of the first compound slide system (19). The second tool holder (46) can be moved in controlled fashion along two axes by a second compound slide system (28, 30, 32, 34). The movement of the second tool holder (46) is synchronized with the movement of the first compound slide system (19) in such a way that the movement of the second tool holder (46) is an arithmetic overlay of the movement of the first compound slide system (19) and the relative movement between the workpiece and the second tool holder (46).