Screw-cutting lathe name "screw-cutting lathe" carries a taxonomic qualification on its use—it is a term of historical classification rather than one of current commercial machine tool terminology. Early lathes, many centuries ago, were not adapted to screw-cutting. Later, from the Late Middle Ages until the early nineteenth century, some lathes were distinguishable as "screw-cutting lathes" because of the screw-cutting ability specially built into them. Since then, most metalworking lathes have this ability built in, but they are not called "screw-cutting lathes" in modern taxonomy. The screw has been known for millennia. Archimedes devised the water screw, a system for raising water. Screws
Archimedes' quadruplets theorem: Also: Combining these gives: Expanding, collecting to one side, and factoring: Solving for "x": Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas. Archimedes' quadruplets In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles. An arbelos is formed from three collinear points "A", "B", and "C", by the three semicircles with diameters "AB", "AC", and "BC". Let the two smaller circles have radii "r" and "r",