Abstract: In an apparatus for machining polygon profiles, comprising a workpiece spindle and a tool spindle (12) driven in synchronism therewith and carrying an eccentric tool (22), wherein the transmission ratio between the workpiece spindle and the tool spindle as well as the lateral distance therebetween are adjustable, the invention provides for additional adjustment of the eccentricity of the tool. This permits one and the same tool to be employed for pre-cutting various polygon profiles of any standard nominal size in a rational manner and with close approach to the final profile shape. The cutting speed is increased by providing a disk-shaped counterweight (21) with adjustable eccentricity located on the tool spindle adjacent the tool. A common adjustment means (18, 24 to 27) is effective to automatically adjust the eccentricity of the counterweight in correspondence to the respective eccentricity of the tool.