Studies for the design of a perpetual wheel

This sheet represents the culmination of Leonardo’s studies on perpetual motion. Examined here are four types of wheel with different mechanical solutions to break the equilibrium. The first is the traditional wheel with articulated arms illustrated by Villard de Honnecourt and Taccola, which here is integrated with an anti-return system consisting of a gear with asymmetrical teeth and a prop, which prevents the wheel from turning in the opposite direction. The study of this wheel is accompanied by a drawing illustrating the articulation of the human arm which, according to Leonardo, provides by analogy the idea for the stop system that keeps the arm in the position of maximum opening.
The second model develops the overbalanced wheel with the hollow semicircular spokes of the Indian tradition. Leonardo’s model consists of twelve semicircular paths each featuring a sphere free to move. The spheres move from the centre of the wheel towards the circumference following a convex path. In this way the wheel is induced to move counterclockwise by the percussion on the rim and by the greater arm of leverage determined by the sphere placed at the maximum distance from the centre. When the spheres begin the passive phase of ascent along the concave path, they move towards the centre, decreasing the lever arm and apparently producing a lesser moment. This overbalanced condition led to the belief that the system could move in perpetual motion.
In the third model the balls follow a diametrically long path and there are four closed circuits mounted on an octagonal hub. Each circuit contains a sphere, which moves along the rectilinear part until it hits the rim on the opposite side, inducing the clockwise rotation of the system. This is probably a model originally devised by Leonardo and developed in an attempt to amplify the effects of the wheel with semi-circular sections. The illustration of a module of these diametric circuits takes us to the heart of Leonardo’s analytical method, which first breaks down complex systems into their constituent elements and then uses these to develop new solutions.
The fourth wheel is a device combining four circuits, each made up of three straight paths and three curved path displaced by 120 degrees. Each circuit has two spheres that move from the side of the direction of rotation towards the outer rim and from the passive side towards the centre. Mounting four of these elements on the same axis results in a rotary system with a sphere falling every 30 degrees of rotation, thus producing a series of pulses which it was thought would create the necessary overbalance to generate rotation. This is the most advanced perpetual wheel system developed by Leonardo. The sheet illustrates the development of the idea through sketches of various devices: they reveal that Leonardo also took into consideration modules with four paths and with radii of the circular sections featuring a different curvature, studied their efficiency with a diagram of the moments involved in the system, and then concluded that the best solution was that with three paths offset by 120 degrees.
The wheels of folio 1062r are developed in an analytical way and drawn in fair copy in Codex Forster II (ff. 90r-92v), where Leonardo concludes that it is impossible to achieve perpetual motion with this type of device.