Features:

    1. The animation sort-of makes it clear why the ball must rise as the screw is rotated by external cranking.

      But to explicate in words, say that we start with the screw at rest and the ball at mid point along the rotation axis of the screw also at rest.

      Note the ball is at a minimum height in the screw relative to the adjacent curvature of the screw which curves upward in the vertical direction. Look closely to see this. We assume the ball is frictionless, and so CANNNOT roll, but only slide.

      Now when the screw rotates counterclockwise (as seen from the upper end of the Archimedes' screw), it tries to the drag the ball uphill counterclockwise and keep the ball at the same point on the screw thread, but the ball slide downhill, but that makes it move upward along the screw thread, and so upward along the direction of rotation axis of the screw.

      Another way of looking at the situation is to say the ball stays in a gravity well perpetually, but the gravity well moves uphill along the direction of rotation axis of the screw.

      Note grade (slope) of the Archimedes' screw must be kept low in order to form a gravity well.

    2. Archimedes' screw in practice is used for raising a fluid (e.g., water for irrigation) or, alternatively, for extracting mechanical energy from a falling fluid. Either way, the fluid is given some horizontal displacement.

    3. Archimedes' screw is an example of applied physics since one needs physics to understand it and modern-day optimization. But Archimedes' screw was certainly invented empirically by people just playing around with screws and pipes perhaps after an observation of an accidental Archimedes' screw.