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 Palimpsest a parabola and one of its secant lines. (For explicit details, see Archimedes' use of infinitesimals.) When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the