The blade (50) has a helicoidal shape with an external cutting edge (50A) having increasing radius and developing over a partial sector, the difference between maximum and minimum radius being at least equal to the maximum diameter of the rolls or logs (B) to be cut. The shaft (45) and the axis of rotation of the blade rotor (42) are not displaced during the rotor's revolution and are slightly inclined with respect to the direction of advancement of the logs within their carriers. Thus the cut performed on the rolls (B) and the cutting edge (50A) will, on the average, be perpendicular to the axes of the rolls (B). The advancement of the roll(s) (B) in the carriers is continuous and the cutting takes place during a portion of each revolution of the rotor and during that portion of the roll's advancement which corresponds to the axial length of the small rolls to be produced.