A grinder for grinding an outer surface of a roll, in particular one of a tissue cylinder and a machine glazed cylinder, the roll having an axis, including a grinding mechanism displaceable in a direction substantially parallel to the axis of the roll, the grinder mechanism also displaceable toward and away from the axis and a measuring system associated and displaceable with the grinding mechanism, the measuring system measuring the position of the grinding mechanism relative to the roll and relative to at least one reference line that is outside of the roll, the grinding mechanism being adjustable substantially parallel to the axis. The measuring system obtaining measurement values, the measurement values being used to positionally control the grinding mechanism.