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Mechanical Paradox Demonstrator

Orrery Picture Orrery Engraving
Click to enlarge the engraving.

James Ferguson's Mechanical Paradox Orrery is based on an unusual gear train. Armstrong Metalcrafts presents a version of this gear train that can be handed around to people to experience the gear train directly. By turning the knob underneath the demonstrator one can see firsthand the differential rotation of the pointers respresenting the nodes and apogee of the Moon's orbit:

(You may download this video as MP4, WebM, or Ogg.)

We're pleased to make this available to people wishing to enhance their presentation of Ferguson's great invention.

The Paradox

A simplified version of the paradox reads:
"Three wheels on the same axis mesh with one thick wheel. Turn the thick wheel. One of the thin wheels goes forward, one backwards, and one goes no way at all!"

What Ferguson was really doing with his Mechanical Paradox was emphasizing the importance of defining “rotation” with respect to a fixed frame of reference.

Ferguson Orrery Gear Layout 1 The “paradox” arises when in a train of gears – A, B, and C – gear A is fixed and gears B and C have epicyclic motion around it. Gear A is the gear under the sun and is fixed to the base. When all three gears have the same number of teeth, gear B rotates twice for each rotation and gear C maintains its orientation to a fixed frame of reference. That keeps the Earth’s axis pointed in the same direction. When gear C has fewer teeth than gears A and B, it turns in the direction opposite the mechanism, in this case illustrating the regression of the nodes. When gear C has a few more teeth, it will slowly turn in the same direction as that of the mechanism, illustrating the advancement of the apogee of the Moon’s orbit.

Paradox Demonstrator Picture

The gears A, B, and bottom C that drive the Earth's axis have 39 teeth. The middle C gear that drives the Nodes of the Moon's orbit has 37 teeth, and the top C gear that drives the apogee pointer has 44 teeth.

Caution: 37 is a Prime Number

When first manufactured, Armstrong Metalcrafts aligns all the pointers to point to the left. The first revolution of the knob shows the differential motion of the gears very clearly. As the knob is rotated through further rotations, the difference in pointer positions will appear to grow and shrink. One might be tempted to think the pointers will soon return to their initial position, but it actually requires 1628 turns. This is because 37 is a prime number, so there must be 37 times 44 turns before the pointers return back to their initial position.


These books have the best descriptions for this orrery. You can find links for these in Armstrong Metalcrafts Bookstore.

Buick, Tony. Orrery 2014; New York, Springer.

Ferguson, James. The description and use of a new machine, called the mechanical paradox; invented by James Ferguson, F.R.S. 1764; London (Reprinted in Select Mechanical Exercises..., 1773; London)

King, Henry C. Geared to the Stars. 1978; Toronto, University of Toronto Press.

Millburn, John R. 1988. Wheelwright of the Heavens. 1988; London, Vade-Mecum Press.

For Sale

Armstrong Metalcrafts is building a limited number of these for sale. The retail price is $295, and this includes a small booklet explaining the history of the mechanical paradox and how it works. To inquire about a purchase, please use our contact form or send an email to "sales" at

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