The current positioning data of a machining tool with regard to a reference surface of a workpiece or of a body connected to the workpiece, is determined. The tool and the workpiece to be machined are relatively moveable toward and away from one another, and the rotary speed of a drive motor for the tool is set to a starting rotary speed which is so low that, when the tool comes into contact with the reference surface, its rotary speed is measurably reduced, the reference surface is moved past the tool before the tool is in contact and while the tool is in contact. As a result, the braking moments in the bearings are increased, and the drive motor comes to a stop quicker.