Study for the design of an overbalanced wheel

The model of the wheel illustrated on this sheet was conceived for the first time in the 11th century by the Indian mathematician Bhāskara, who designed a wheel with semicircular hollow spokes containing mercury. The geometric scheme of the semicircular spokes described two asymmetrical paths: convex during the descent and concave during ascent. This device is also present in the manuscripts of Taccola and Francesco di Giorgio; Leonardo revisited it by replacing the mercury with metal spheres. The same wheel is described in detail in folio 91v of Codex Forster II, however there Leonardo states that it cannot achieve perpetual movement.
The study on folio 44v of the Codex Arundel seems to constitute an attempt at geometric analysis in order to understand the advantage in terms of leverage offered by this configuration of the wheel. Leonardo divides the circumference into circular sections out of sync by 30 degrees and observes that the spheres, by the action of gravity, fall outwards, ending up themselves in the lower part of the path, and so developing a lever arm of equal length to the spoke of the wheel. In this way they would produce a break in the equilibrium in the direction of rotation, since on the upward side, thanks to the semicircular path with a diameter equal to the spoke of the wheel, the same sphere is positioned in such a way as to have a lever arm equal to half the spoke of the wheel. The fact that Leonardo designed a hub with the spheres arranged at the end of the ascent leads us to believe that he was envisaging a real model. The written note is incomplete, so we don’t know the conclusion of his reasoning, but it is clear from the drawing that Leonardo was looking at the conditions that cause the overbalance of the wheel. In fact, he draws each sphere in the position dictated by the centres of gravity and, above, a partial linear development of the semicircle, which seems to be the geometric representation of the intensity of the forces acting on the wheel, following the scholastic tradition.