To provide a band sawing machine which can saw a material which is difficult to be cut such as the material or the like made of a stone without generating a large operating force.

The band sawing machine comprises a workpiece bed 2 which can fix at least one workpiece 1, and a saw blade frame for a band saw blade 3 which moves to circulate. The workpiece 1 and the saw blade 3 perform a relative translational motion and an oscillating motion around an oscillating axial center perpendicular to the cutting plane. The translational motion and the oscillating motion are interlocked and controlled so as to form a curved cutting line on the workpiece, in such a manner that the geometrical parameter of the curve can be adjusted steplessly.