The machining of drills (1) includes an operation for chamfering the cutting edges (8) performed in two steps. In the first step the rotating grindstone (15) is directed to various locations of the cutting edge (8) and the point of contact is detected by an electronic appliance (17) whereof the electronic module (19) controls the stoppage of the grindstone, its return to the starting position and the recording of the position of the grindstone at the time of contact. In the second step the grindstone (15) is controlled to follow the left line obtained by connecting the points registered during the first step.