{
 "version": "2026-10-20",
 "license": "Dimension sheets: CC BY 4.0. Standard and published-size files: CC BY 4.0. Xem Studio's own designs (files_license = Xem Studio Design License): free to use and sell things from, not to redistribute as files — https://xemstudio.com/license/. Other publishers' files: their own license, linked beside them.",
 "slug": "fergusons-paradox-mechanism",
 "name": "Ferguson's mechanical paradox (19 / 20 / 21 tooth wheels on one pin)",
 "kind": "hardware",
 "category": "Mechanisms",
 "group": "Gears",
 "year": null,
 "question": "Why do the three wheels of Ferguson's paradox turn differently?",
 "tags": [
  "mechanism",
  "gears",
  "kinematics",
  "507 mechanical movements"
 ],
 "aliases": [
  "ferguson's paradox",
  "mechanical paradox",
  "paradox gear train",
  "epicyclic paradox",
  "paradoja mecánica de Ferguson",
  "Fergusons mechanisches Paradoxon"
 ],
 "overall_mm": {
  "length": null,
  "width": null,
  "height": null
 },
 "official_cad": false,
 "xem_status": "free-files",
 "les": false,
 "free_files": 1,
 "url": "https://xemstudio.com/objects/fergusons-paradox-mechanism/",
 "sheet_txt": "https://xemstudio.com/objects/fergusons-paradox-mechanism/sheet.txt",
 "sheet": {
  "basis": "Xem Studio design following Brown No. 504 (1868)",
  "overall_mm": {
   "length": null,
   "width": null,
   "height": null
  },
  "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."
  ],
  "table": {
   "columns": [
    "parameter",
    "value"
   ],
   "rows": [
    [
     "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"
    ]
   ]
  }
 },
 "sheet_text": "Ferguson's mechanical paradox (19 / 20 / 21 tooth wheels on one pin) — dimension sheet\nSource: Xem Studio, https://xemstudio.com/objects/fergusons-paradox-mechanism/  (sheet CC BY 4.0; files Xem Studio Design License, https://xemstudio.com/license/)\nBasis: Xem Studio design following Brown No. 504 (1868)\nCatalog version: 2026-10-20\n\nSizes: see table below\n- 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.\n- 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.\n- Brown's No. 504 (Ferguson's mechanical paradox, 1764): the demonstration behind his orrery.\n- 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.\n- Brown's movement No. 504 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.\n\nparameter  value\nsun (fixed)  20 T, module 1\nidler  30 T × 26 face\nwheels  19 / 20 / 21 T on one pin\ncentre distances  25 + 25 mm\nturns per arm turn  −1/19, 0, +1/21\n\nFiles elsewhere (not hosted by Xem Studio; license as stated by the publisher):\n- 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\n- 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\n\nFree files (Xem Studio Design License, Xem Studio; STEP + STL, 1 parts): https://xemstudio.com/files/hardware/fergusons-paradox-mechanism/index.json\nAPI: https://xemstudio.com/api/geometry?object=fergusons-paradox-mechanism\n",
 "sources": [
  {
   "publisher": "Henry T. Brown (1868, public domain)",
   "title": "Five Hundred and Seven Mechanical Movements — scan on archive.org",
   "url": "https://archive.org/details/fivehundredseven00browiala",
   "formats": [
    "web"
   ],
   "license": "Publisher's page; dimensions are facts",
   "license_verified": true,
   "hosted": false
  },
  {
   "publisher": "507movements.com",
   "title": "Brown's movement No. 504: figure, his text and an animation",
   "url": "https://507movements.com/mm_504.html",
   "formats": [
    "web"
   ],
   "license": "Publisher's page; dimensions are facts",
   "license_verified": true,
   "hosted": false
  }
 ],
 "xem": {
  "status": "free-files",
  "sheet_free": true,
  "mesh_price_usd": null,
  "step_price_usd": null,
  "tolerance_mm": null
 },
 "related": [
  "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"
 ],
 "fits": [],
 "dimensions": "exact (nominal, no printing tolerance)",
 "dimension_rule": "All geometry on Xem Studio is exact (nominal, true size) with no printing tolerance. Where a part meets another part, the object lists its mating features (feature, exact size, what it mates with, fit class: press, slide, run, socket, clamp or designed). Apply your process's allowance to those features only; 'designed' gaps are part of the design and stay as drawn.",
 "mating": [],
 "faq": [
  {
   "q": "Why do the three wheels of Ferguson's paradox turn differently?",
   "a": "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."
  },
  {
   "q": "What is the rule for a wheel on the arm?",
   "a": "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."
  }
 ],
 "files_license": {
  "name": "Xem Studio Design License",
  "holder": "Xem Studio",
  "url": "https://xemstudio.com/license/",
  "note": "Xem Studio's own design: free to use, print, modify and sell things made from it; not to be republished or redistributed as files."
 },
 "offers": [
  {
   "file": "fergusons-paradox-mechanism-assembly",
   "formats": [
    "step",
    "stl"
   ],
   "price_usd": 0,
   "license": "Xem Studio Design License",
   "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",
   "note": "Assembled mechanism, all parts placed at one pose (STEP keeps each part separate)"
  }
 ],
 "markdown": "https://xemstudio.com/objects/fergusons-paradox-mechanism/index.md"
}