A cutting device for a cutting machine comprises a chain guided in translation along a cutting plane by a chain guide and has cutting tools arranged alternately on either side of the cutting plane. Each cutting tool has a curved cutting end, which curves towards the outside of the cutting plane and defines a concave surface. A scraper is positioned on either side and in the vicinity of the chain guide and cutting tools parallel to the cutting plane, said scraper having two ends projecting towards said plane, each having a convex surface complementary to the concave surface, so that said ends remove cutting waste from the tools during the passage of the tools on the scraper. The cutting device is especially useful for cutting soft, sticking, or dirtying materials.