# Ferguson's mechanical paradox (19 / 20 / 21 tooth wheels on one pin)

> Why do the three wheels of Ferguson's paradox turn differently?

- Category: Mechanisms › Gears
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 504 (1868)
- Page: https://xemstudio.com/objects/fergusons-paradox-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=fergusons-paradox-mechanism · Plain text: https://xemstudio.com/objects/fergusons-paradox-mechanism/sheet.txt
- License: dimension sheet CC BY 4.0 (Xem Studio); files Xem Studio Design License (free to use and sell from, not to redistribute as files: https://xemstudio.com/license/)
- Use in every chat: add https://xemstudio.com/mcp as a connector (https://xemstudio.com/use-with-ai/)

**Exact dimensions.** True size, with no printing tolerance added.

## Notes

- A 20-tooth sun is fixed; an arm turning about it carries a thick idler and, on a second pin, three loose wheels of 19, 20 and 21 teeth that all mesh with the idler. Carry the arm once round: the 20 T wheel does not turn at all, the 21 T wheel turns 1/21 of a turn forwards and the 19 T wheel 1/19 of a turn backwards — each wheel turns (1 − 20/N) turns per turn of the arm, the idler only setting the direction.
- Three wheels on one pin at one centre distance (25 mm) with different tooth counts need profile shifts: +0.54 m on the 19 T and -0.46 m on the 21 T, 20° involutes meshing an unshifted 30 T idler (tips shortened to Ø31.5) with no backlash drawn; the 20 T wheels are catalog gears.
- Brown's No. 504 (Ferguson's mechanical paradox, 1764): the demonstration behind his orrery.
- Parts (each has its own page): spur-gears-module-1, spur-gear-m1-30t-w26-b6, spur-gear-m1-19t-x054-b6, spur-gear-m1-21t-xm046-b6, paradox-arm-25-25-b6, dowel-pins-metric.
- Brown's movement No. 504 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| sun (fixed) | 20 T, module 1 |
| idler | 30 T × 26 face |
| wheels | 19 / 20 / 21 T on one pin |
| centre distances | 25 + 25 mm |
| turns per arm turn | −1/19, 0, +1/21 |

## Where the files are

- Henry T. Brown (1868, public domain): [Five Hundred and Seven Mechanical Movements — scan on archive.org](https://archive.org/details/fivehundredseven00browiala) — web — Publisher's page; dimensions are facts
- 507movements.com: [Brown's movement No. 504: figure, his text and an animation](https://507movements.com/mm_504.html) — web — Publisher's page; dimensions are facts

## Free STEP/STL (1 parts, Xem Studio Design License, Xem Studio)

Index: https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/index.json · license: https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/LICENSE.md

- fergusons-paradox-mechanism-assembly: [STEP](https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/fergusons-paradox-mechanism-assembly.step) · [STL](https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/fergusons-paradox-mechanism-assembly.stl) — Assembled mechanism, all parts placed at one pose (STEP keeps each part separate)

## Questions

**Why do the three wheels of Ferguson's paradox turn differently?**

A 20-tooth sun is fixed; an arm turning about it carries a thick idler and, on a second pin, three loose wheels of 19, 20 and 21 teeth that all mesh with the idler. Carry the arm once round: the 20 T wheel does not turn at all, the 21 T wheel turns 1/21 of a turn forwards and the 19 T wheel 1/19 of a turn backwards — each wheel turns (1 − 20/N) turns per turn of the arm, the idler only setting the direction.

**What is the rule for a wheel on the arm?**

Relative to the frame it turns 1 − N_sun/N per turn of the arm (idler ignored): 0 for 20 T, +1/21 and −1/19 here.

## Fits with

- [Spur gears, module 1: 10 to 80 teeth (20° involute)](https://xemstudio.com/objects/spur-gears-module-1/)
- [Spur gear, module 1, 30 T, 26 mm face (thick idler), Ø6 bore, tips at Ø31.5](https://xemstudio.com/objects/spur-gear-m1-30t-w26-b6/)
- [Spur gear, module 1, 19 T, profile shift +0.54, Ø6 bore (meshes a 30 T at 25 mm)](https://xemstudio.com/objects/spur-gear-m1-19t-x054-b6/)
- [Spur gear, module 1, 21 T, profile shift -0.46, Ø6 bore (meshes a 30 T at 25 mm)](https://xemstudio.com/objects/spur-gear-m1-21t-xm046-b6/)
- [Train arm 64 × 14 × 5 with three Ø6 holes at 25 mm (Ferguson's paradox)](https://xemstudio.com/objects/paradox-arm-25-25-b6/)
- [Dowel pins, 2–6 mm (ISO 2338 / DIN 6325)](https://xemstudio.com/objects/dowel-pins-metric/)