In an apparatus for use in machining a tooth system of straight or helically toothed gears on a workpiece, said apparatus including a hyperboloidally or globoidally formed toothed tool having an abrasive surface on the teeth thereof and which has a sufficient axial dimension such that it meshingly engages said workpiece tooth system from one axially facing side thereof to the other and with normal backlash between the teeth thereof, first means defining a first crossed-axes angle relation between said workpiece and said tool, a first guide gear and first connecting means for coupling said first guide gear to said tool and being rotatable therewith, a second guide gear and second connecting means for coupling said second guide gear to said workpiece and being rotatable therewith, said first and second guide gears operatively engaging each other and having the same speed ratio as between said workpiece and said tool, said first and second guide gears operatively engaging each other and having normal backlash between the teeth thereof, and control means for controlling the relative positions between teeth on said workpiece and said tool and teeth on said first and second guide gears so that the operatively engaged combination of said workpiece/tool and said first guide gear/second guide gear is entirely backlash free, the improvement comprising wherein second means defines a second crossed-axes angle relation between said first guide gear and said second guide gear which, when meshingly engaged, is independent of said first crossed-axes angle relation.