Band saw apparatus (21) is disclosed. It includes: a drive assembly (22, 27, 28) formed for support and guided movement of a saw blade (29) for cutting of a workpiece (81); an endless band saw blade (29) mounted to the drive assembly; and reciprocation apparatus (33, 34, 41) formed for reciprocation of the blade (29) toward and away from the workpiece (81) to enable see-sawing of the blade (29) and variation of the force per unit area of the cutting edge (66) of the blade (29) on the workpiece(81) white maintaining the same total force on the blade (29). To enhance efficiency of cutting, the blade drive assembly is mounted to the remainder of the saw (21) to be substantially free to rebound or move away from the workpiece (81) during reciprocation to avoid digging in or shearing off of the blade teeth and to permit close spacing of the blade guides (33, 34) next to the workpiece (81). The saw (21) includes a control circuit (111, 154) for controlling the rate of movement of the saw arm (22) away from (36) and toward,(37) the workpiece (81) to enable timing of saw arm (22) rebound to blade (29) displacement. The blade guides (33, 34) are reciprocated by the fluid piston-cylinder assemblies (41) in synchronism or out-of-phase to vary the amplitude, frequency, speed and dwell during displacement independently of blade (29) speed and at two locations along a length of the blade (29) so as to produce see-sawing oscillation as well as simple reciprocation. The blade guide assemblies (33, 34) are provided with displacement means (41) to permit guide assembly (33, 34) adjustment during cutting to maintain the guide assemblies (33, 34) in close proximity to the workpiece (81). Alternative blade displacement assembly constructions and a method for cutting workpieces employing the apparatus also are disclosed. The method includes variation of the feed rate of the saw blade (29) down through the workpiece (81) as a function of the width of the workpiece (81).