Connecting from the mechanism to the face:

There is a shaft connected to the driving mechanism via a universal. The purpose of the universal is to allow the shaft to be connected to and driven by the main mechanism without needing to be perfectly aligned with the turning axis of the mechanism. For instance, if the center of the face is situated much higher than the driving mechanism, then the shaft leading from the mechanism to the face could not possibly be parallel to that mechanism’s axis of rotation and on the same elevation. A shaft connected to the universal that isn’t perpendicular to the plane in which the mechanism is rotate will be caused to rotate at a rate different from that of the mechanism, however, this can be rectified by means of another universal attached to the other end of the shaft (at the clock face), such that the final axis of rotation is parallel to the original axis of rotation.

We want the two hands to turn about the same axis, so we need the gears to be positioned such that we have two shafts that rotate about the same center, but at different rates. More specifically, the ratio of angular velocities of the hour hand to the minute hand should be 1:12.

The 4 gears A, B, C, D have 15, 45, 12 and 48 teeth respectively. Gears B and C are attached to each other such that they turn about the same axis and at the same rate. So, when gear A (which is attached to the universal) turns clockwise, the combination of gears B and C turn counterclockwise at a third the rate and gear D turns clockwise at a fourth the rate of B and C, and hence a twelfth the rate of the gear A.

Gearing:

The gearing mechanism above makes use of involute gear. Below is a comprehensive description and animation of the involute gear from Wikipedia:

The involute gear profile is the most commonly used system for gearing today. In an involute gear, the profiles of the teeth are involutes of a circle. (The involute of a circle is the spiraling curve traced by the end of an imaginary taut string unwinding itself from that stationary circle.)

In involute gear design, all contact between two gears occurs in the same fixed, flat plane (the Plane of Action), even as their teeth mesh in and out. Further, the contacting surfaces are always perpendicular to the plane of contact, so the dominant contact forces (in a well lubricated system) are always parallel to the plane. This way the moment arms are kept constant. This is key to minimizing the torque/speed variations which produce vibration and noise in lower quality gears. Note that the involute profile does not prevent the teeth from scraping each other every time they mesh, and this is the dominant source of wear. It is not possible to design a gear tooth profile which rolls through the mesh without friction. Service life is often managed by using hard materials and constant lubrication. When friction wear is a critical issue, chain drives can help reduce maintenance requirements.

The involute profile can be generated using a hobbing machine with a rack form. Rack pressure angles of any degree are theoretically possible, however only 3 have been in common use; 14.5 degrees which was used many years ago but now is obsolete, 20 degrees which is the most common and 25 degrees which is generally only found in the USA.

Small pinions have addendum modifications to stop interference.

Source: http://en.wikipedia.org/wiki/Involute_gear