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Ferguson's mechanical paradox (19 / 20 / 21 tooth wheels on one pin)
Why do the three wheels of Ferguson's paradox turn differently? · Hardware · basis: Xem Studio design following Brown No. 504 (1868)
Also called: ferguson's paradox, mechanical paradox, paradox gear train, epicyclic paradox, paradoja mecánica de Ferguson, Fergusons mechanisches Paradoxon
Exact dimensions. True size, with no printing tolerance added.
- sizes
- 5 in table
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.
Ferguson's mechanical paradox size chart
| 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
Other publishers' files are linked, not hosted. "verify" = check the license at the link before commercial use.
| Henry T. Brown (1868, public domain) | Five Hundred and Seven Mechanical Movements — scan on archive.org | web | Publisher's page; dimensions are facts |
| 507movements.com | Brown's movement No. 504: figure, his text and an animation | web | Publisher's page; dimensions are facts |
Free files
★ Free files 1 parts. Xem Studio design: free to use, print and sell things from; not to be republished as files. Xem Studio Design License. index.json
| fergusons-paradox-mechanism-assembly | Assembled mechanism, all parts placed at one pose (STEP keeps each part separate) | STEP · STL |
https://xemstudio.com/mcp to Claude, ChatGPT, Cursor or any MCP client (free, no account, one minute) and it can look up exact sizes, mating fits and free STEP/STL from every chat.Plain text: sheet.txt · JSON: /api/geometry?object=fergusons-paradox-mechanism
Fits with
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.