Gears, shafts and cams

This lesson asks you to describe applications of simple machines, and to explain how compound machines operate and what mechanical advantage they give. The museum offers three machines described in medieval texts: a calendar gear train, a water-driven pump, and a pump with six pistons and cams; the levers, pulleys, calculations of mechanical advantage and biomechanical systems are for your class. ¹

Look at each machine and list its parts. Mark the parts that share a shaft, the parts that mesh with another part, and the parts that push something along a line. What does each connection do to the speed or the direction of the motion?

Count the teeth. Al-Biruni described an older device called the Moon Box, then proposed simplifying its solar gearing: a four-tooth gear driving a fifty-two-tooth gear ². In the proposal the four-tooth gear shares the forty-tooth lunar wheel's arbor, so they turn together; the four then meshes with the fifty-two ³. A shared arbor transfers the same rotation, whereas a mesh changes speed according to tooth counts ³. Four teeth driving fifty-two make the solar pointer's revolution thirteen times longer than the sidereal lunar revolution ⁴. The user advances an input by hand, and a complete input revolution takes a week ⁵.

The record keeps no sentence in which al-Biruni says why he proposed the change. It does give Hill's suggestion that reversibility might help investigate dates, including debt dates. The record calls this a possible interpretation of usefulness, not a transaction archive proving adoption by a court or market ⁶.

Fewer wheels do not make a better calendar. Hill's reconstruction of the older design gives about 366.4211 days for the solar pointer, which is about 1.1789 days too long against his 365.2422-day reference year, while the 364-day proposal is 1.2422 days short ⁷. So the proposal is not more accurate than the older arrangement on every measure: it is an attributed simplification, not an unconditional mathematical victory ⁷.

Hill derives a 364-day year from the ratios, whereas the translated passage mentions 360 days, and the difference remains visible in the record ⁸. Hill also warns of erroneous drawings and partly inferred assembly, so tooth counts alone cannot rebuild the machine ⁹. The record read Hill's English introduction, translation and commentary, not the complete Arabic apparatus ¹⁰.

Hill's translation says 360 days, and his reconstruction from the gear teeth gives 364. Which would you trust, the number written in the passage or the number the gears give, and why?

• Al-Biruni's calendar gearing: the eight wheels, the proposal and the two numbers.
• The double-piston pump: another way to turn rotation into another motion.
• The six-cylinder pump: cams, rockers and weights around one shaft.
• Al-Biruni: the scholar and the disagreements about his affiliation.

Read about al-Biruni's calendar gearing, the double-piston pump and the six-cylinder pump. Draw one of them as a chain of parts, and label each link as a shaft, a mesh, a pin in a slot or a cam.