Archimedes' screw the outer casing to turn the entire apparatus as one piece, which would require that the casing be rigidly attached to the screw. Some researchers have postulated this as being the device used to irrigate the Hanging Gardens of Babylon, one of the Seven Wonders of the Ancient World. A cuneiform inscription of Assyrian King Sennacherib (704–681 BC) has been interpreted by Stephanie Dalley to describe casting water screws in bronze some 350 years earlier. This is consistent with classical author Strabo, who describes the Hanging Gardens as irrigated by screws. German engineer Konrad Kyeser equipped the Archimedes screw with
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