Abstract: Transversal cutting machine comprises feeding means (20) that supply a log of paper (15), for example hygienic paper, to a sawing station at which rolls or stacks (16) are cut off from the log (15) by a blade (25). During the cutting step the log (15) is held by a clamping device (1), divided in two parts (2, 3) arranged respectively upstream from and downstream of a slit (17) through which the blade (25) passes. In particular, the first part (2) and the second part (3) of clamping device (1) are movable relatively so that during the cut the log (15) is locally subject to traction at the same time of the blade (25) driven in the log. This way, during the cutting action of the blade (25) an opening (17) is created on the log (15) having a size at least the same as the thickness of blade (25) same. This makes easier the action of blade (25) since it reduces the friction produced between the blade (25) and the log (15).