Leonardo called the effect of these two machines “contrary friction”. The first, operated by the handle, makes use of the movement of a large toothed wheel with pegs arranged in groups on the flat surface. These groups of pegs engage alternately with, first, the small cylinder at the top and then with the one at the bottom. The two cylinders then transmit the reciprocating motion to the toothed, 3-sided curved part (lunula), which makes the notched bar slide first one way, then the other.

The second idea is simpler: the handle turns two toothed semi-circles and these engage alternately with the pegs on the two adjacent bars. The two bars are linked by a cord passed around the cylinder, which returns them to their original position, at the same time receiving reciprocating motion from them.

On page 7r, Leonardo sets out the rule on the number of teeth in a gear. He also portrays two machines to create reciprocating motion. The first, moved by the handle, turns a wheel with 5 pegs or teeth that move alternately above and below two blades connected to a vertical rod. The rod with the blades moves alternately, moving the horizontal rod by reciprocating motion. The second machine shows two large wheels moved by a handle. Each wheel has 9 long pegs staggered against those on the opposite wheel so that they engage and push the split lever alternately, making it move from side to side. Finally, the mechanism pushes the upper rod with reciprocating direct motion.

These two systems include a remarkable innovation: a wavy line is carved into a large wheel, making a track that can be “programmed” as desired. One or more pointed rods fixed to a pivot are inserted into the groove so that the pivot has to follow the track when the wheel is turned. In this way, oscillating motion is produced which can also be programmed by altering how the wheel turns. In the first example, the symmetrical twin tracks could be used to operate a pair of shears held in position by a block above the wheel. In the second, the mechanism is a blade like the ones used in clocks, but Leonardo says it is quieter.

This system of rings enables the inner hemisphere to keep its original position independently of its rotation. Two ring pivots fix the three outer rings to each other as they rotate, with a 90° displacement between each pair of rings. In this way, the inner system can move freely on three axes (X, Y, Z). The weight beneath the rotational axis keeps the inner hemisphere horizontal. The same system had always been used on ships to hold oil lamps steady in spite of the pitching caused by waves.

Inside the cylinder there is a spring that moves the toothed wheel below. On top, there is a metal arch with a castor that runs and rests on the upper surface of the cylinder, which has a step in it. The metal arch keeps the castor under pressure. In this way, the mechanism can only turn in one direction, because if it turned the other way it would be blocked by the castor bumping into the step.

This complex gear system makes maximum use of the energy produced by a spring. The tightened spring is hidden inside the container. When the stop lever is released, the spring begins to turn the helical (spiral) gear, which makes use of the initial strong energy from the spring to rotate briefly. As the gear rotates, it automatically moves down the screw, increasing its pace as it descends. In this way, full use is made of even the spring’s slight residual energy. The helical gear turns the cylinder and thus the upper wheel. At the same time, the lower mechanism enables the spring-blocking device to move slowly and gradually to the right. The handle is used to rewind the spring by hand.

The tightened spring is hidden inside the container; its center of rotation is the axle. Winding is therefore decentralized and will cause the drum to move in an odd, but useful, way. In fact, a toothed spiral is placed on the drum which rises as it becomes further away from the axle. As it rotates, this spiral gear engages the upper conical cylinder which is fixed between points n and h. The cylinder is a conical cage-wheel gear. The narrow radius makes use of the initial energy from the spring, while the wide radius at the end exploits the residual energy. The cylinder is connected directly to the final wheel.

Leonardo tried to improve the system of reciprocating motion because it had gear problems. Here he suggests making the main mechanism by using a quarter-circle (45° segment). The handle turns the wheel which has 32 teeth arranged on only half its circumference. The teeth engage alternately first segment, which makes disc rotate, completing four turns. At the next stage, wheel, which continues turning in the same direction, engages segment which, in the same way as above, makes disc rotate four times. Interestingly, Leonardo uses a bell which is shaken when segment reaches the end of its course and rings while segment is turning.

In this system, transmission of the manually driven reciprocating motion to a bell that rings is made by means of cloth or leather belts. Leonardo suggests using belt transmission to avoid the noise of the gears. In the second mechanism the reciprocating motion is supplied by the double-headed toothed “axe” which engages alternately with the cylinders. The “axe” is operated manually by the rod.

By turning wheel, the cogwheel moves the little disc, which can only move along the straight groove. If it is fixed or the assembly is stood on end, gravity will make it follow the same pre-programmed path back and forth. On wheel there is a groove in the shape of a double spiral. In this case, a spool is inserted in the groove. When the wheel turns, the spool is fixed so that it can only move in one direction, thus it has to follow the pre-programmed course; because it is lens-shaped it can even negotiate the points where the tracks cross. By altering the shape of the groove, it is possible to program the movement in the desired direction.

The system is operated by the handle which transmits motion to the wheel via an endless screw. The wheel always turns counterclockwise. Wheel R1 has 12 teeth arranged only at the front, with the result that the sequence is divided into two movements. First the wheel engages the bottom cylinder, which pushes the belt and makes it move clockwise. Then, alternately at each turn of wheel R1, the top cylinder is also engaged and this makes the belt move counterclockwise. So the belt moves forward and backward alternately, carrying along the iron rod attached to it. Leonardo suggests that the handle should not be turned too quickly, otherwise the gears jump out of place.

This spring-powered motor is a development of two motors on pages 4r and 16r. The spring, which this time Leonardo states must be of tempered steel, supplies rotary movement to the whole of the central block. In this way, the cylinder, which is fixed to run only vertically between the central axle and the lateral axle, is pushed and made to rotate over the toothed “spiral staircase”. The teeth of the cage gear support and engage the teeth on the outside of the spiral, whereas point C rests on the smooth inner part of the spiral. In two and a half turns this system makes (the others only allowed one turn), the cylinder which is connected to a large wheel with a ring gear not only to rotate, but move upwards, engaging the top disc. The gear turns and slides on the grooved cylinder. To illustrate the mechanism clearly, Leonardo has also shown a section through the central motor.

Very often Leonardo examines the mechanisms and suggests various experiments with minor modifications in order to find better solutions. In this case, for example, he analyzes the efficiency of the rod-and-handle system, suggesting two types of connecting rods (which he calls “la mezana”), one short and one long. He then suggests using an extremely long rod, which has a smoother movement, instead of a short one that may even hamper it.

To operate a flywheel, an inverse rod-and-handle system is used, and this later became the principal mechanism of the steam engine. In this case, to increase speed, Leonardo suggests doubling, tripling and quadrupling the handle assemblies. In the system with four handles set around the flywheel, he adds two rotating discs that indicate a use for the extra energy obtained. The system with four rods is mechanically similar to modern engines with 4-pistons connected to four rods that turn the axle.

A pulley can also be used to move gears, not just to lift weights. In addition, there are countless possible ways of combining different types of motion and the length of the rope allows the movement to be transmitted over a very long distance. The friction and noise produced by gears are also eliminated. It is possible to obtain rotation in any direction, depending on the way the rope is arranged and the inclination of the pulley. What is essential is that the rope must be turned around at least half of the pulley wheel so that the friction will engage it. These systems are the basis for the robot soldier.

In experimenting with the use of pulleys, Leonardo also suggests methods to test his theories. In this case, there are five pairs of weights, each pair connected by a rope and pulleys. The rope for each pair (A, B, C, D and E) passes by way of the central, rotating pillar and, in descending order, each rope is given one more half-turn around the pillar than the previous one. The aim is to understand how far the friction from the pulleys can move the weights as the rope is pulled. In fact, the Codex Madrid is also full of ideas on statics and geometry explained by means of mechanical experiments.

Leonardo also studied tools for making his machines. In this case, he suggests a drill with a diamond-tipped bit, which could be used to make holes in any kind of material. The power comes from a large, handle-operated flywheel whose inertia keeps the system turning. The flywheel makes the drill cylinder rotate. A large ball of lead sits on top of the drill to exert pressure on the piece being drilled. The diamond tip must be cooled with water - exactly the same process used in today’s industry.

This is one of the many mechanisms which can be considered a machine in itself, because we can see its use. In fact, other than the simple machines and composite mechanisms, Madrid I includes suggestions for several machines that are almost complete and ready to work. In this case, Leonardo studies a mechanism for polishing stone or mirrors. In the first system, the handle engages the axle underneath, rotating the plate which holds the mirror to be polished. At the same time, the handle moves a system of connecting rods which supplies rectilinear reciprocating motion to the polishing stone. The stone rests directly on the mirror and runs between four vertical rollers. In the second system, the handle engages the wheel directly and at the same time makes the system of three rods joined at point X move backwards and forwards.