Ferguson's mechanical paradox (19 / 20 / 21 tooth wheels on one pin) — dimension sheet
Source: Xem Studio, https://xemstudio.com/objects/fergusons-paradox-mechanism/  (sheet CC BY 4.0; files Xem Studio Design License, https://xemstudio.com/license/)
Basis: Xem Studio design following Brown No. 504 (1868)
Catalog version: 2026-10-20

Sizes: see table below
- 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.

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

Files elsewhere (not hosted by Xem Studio; license as stated by the publisher):
- Henry T. Brown (1868, public domain): Five Hundred and Seven Mechanical Movements — scan on archive.org — web — Publisher's page; dimensions are facts — https://archive.org/details/fivehundredseven00browiala
- 507movements.com: Brown's movement No. 504: figure, his text and an animation — web — Publisher's page; dimensions are facts — https://507movements.com/mm_504.html

Free files (Xem Studio Design License, Xem Studio; STEP + STL, 1 parts): https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/index.json
API: https://xemstudio.com/api/geometry?object=fergusons-paradox-mechanism