Archimedes' screw screws of decreasing pitch are used for the compaction of waste material. If water is fed into the top of an Archimedes screw, it will force the screw to rotate. The rotating shaft can then be used to drive an electric generator. Such an installation has the same benefits as using the screw for pumping: the ability to handle very dirty water and widely varying rates of flow at high efficiency. Settle Hydro and Torrs Hydro are two reverse screw micro hydro schemes operating in England. The screw works well as a generator at low heads, commonly found in English
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