A method and a device for forming cut surfaces in a transparent material, particularly in the cornea, by producing optical breakthroughs in the material by application of laser radiation focused into the material. The focal point is adjusted in three dimensions to form the cut surface by the sequential arrangement of optical perforations. The focal point is guided in such a manner that cutting is divided into at least two steps, and in at least one of the steps, the formation of the cut is carried out with a path radius that decreases in size, and in one of the steps, the cut formation is carried out with a path radius that increases in size.