Stage 4

We jump now to stage 4b, which is the drive for the moon and for its phases. We will come back to stage 4a, but for now it is easier to understand this section if we regard wheel 24 as being static, as if it were bolted to the fame. The purpose of this section is to make the moon travel anticlockwise round the sky so that in one Synodic month it returns to its starting position with respect to the Sun; e.g. from Full Moon to the next Full Moon.
	A bracket supporting a transported gear train is mounted on the face of wheel 160. Part of this train is an 8-tooth pinion which meshes with wheel 118. In the fashion that we have become used to, wheel 118 is dragged around with wheel 160 and its bracket. At the same time wheel 12 is carried around “static” wheel 24. Wheel 12 is rotated by this, as is the worm. The latter drives an 8 tooth pinion which is connected to the one meshed with wheel 118.
	As wheel 118 is driven round, the pinion driving it slowly rotates such that when wheel 160 has made 29.5 turns, wheel 118 has made one turn less. Thus, wherever the moon started, after 29.5 days is has returned to its original position with respect to the sun.
Each day wheel 12 is taken once round “stationary” wheel 24, and so makes 24/12 turns. The worm, which has an equivalent of 2 teeth, does the same and so the pinions 8 make 24/12 x 2/8 turns. Wheel 118 therefore is retarded by 24/12 x 2/8 x 8/118 = 0.0338983 turns per day and so makes 1 - 0.0339 = 0.9661017 turns. After 29.5 days the moon has made 29.5 x 0.9661017 = 28.5 turns.
But what about the 44 minutes? I hear you cry. Yes, of course the synodic month is 29.5 days and 44 minutes. Richard included a whole section to take care of this - stage 4a. In stage 4b we imagined wheel 24 to be immovable; but this is not the case. In fact it rotates very slowly indeed to provide the extra motion to account for the 44 minutes.
Wheel 12A is of a very unusual design; it is rather as if the spiral teeth had been taken from a worm and laid on a flat wheel. Wheel 12* engages with this wheel such that each solar day, when wheel 12A makes one revolution, wheel 12* also turns once. The reason Richard chose such an unusual design to create a 1:1 drive is a mystery. The motion is transmitted via the worm, wheel 59 and pinion 6 to wheel 196 which is fixed to wheel 24. The latter makes roughly 0.001 turns per day, so actually takes nearly 3 years to make one complete revolution.
The worm has the equivalent of 2 teeth, so every day wheel 59 makes 2/59 turns, as does pinion 6. Hence wheel 196 makes 2/59 x 6/196 = 0.0010377032 turn, as does wheel 24. Now, we previously assumed that when wheel 160 made 1 turn, wheel 12 would travel once around wheel 24, which was stationary. But we now know that wheel 24 is not stationary, but has made 0.0010377032 turn backwards, which means that wheel 12 makes the equivalent of 0.998962296 turns around wheel 24. This in turn means that wheel 12 makes 1.997924594 turns. If we substitute this into the formula above 1.997924594 x 2/8 x 8/118 turns = .033863128 Every day the moon’s wheel makes 1- 0.33863128 turns = 0.96613687 turns. So after 29.53055 days the moon’s wheel makes 29.53055 x 0.96613687 = 28.53055 turns, which is exactly one turn less than the sun. Without the correction provided by the wheel of 44 minutes, the moon’s wheel would have made only 28.5295 turns.

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