Epitrochoid, Kinematic Model by Martin Schilling, series 24, model 1, number 329

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the first in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.
Circles rolling around the outside of other circles, known as epicycles to the ancient Greeks, were used to describe the motions of the planets in a geocentric cosmology. These curves,called epitrochoids, are formed by tracing a point on the radius or the extension of the radius of a circle as it rolls around the outside of a second stationary circle.
Epitrochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and hypotrochoids (see items 1982.0795.02 and 1982.0795.03). In the 18th century, it was found that when shaping the sides of gear teeth as the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.
Depending on the distance of the tracing point from the center of the rolling circle, an infinite number of curves can be formed. The three curves depicted in this model are bicyclic, meaning the smaller circle needs to rotate around the larger circle twice before returning to is original configuration. The ratio of the radii of the two circles will determine the number of nodes in the curve and how many rotations are required before the tracing point returns to its starting configuration. The Spirograph toy produces various types of epitrochoids. (See item 2005.0055.02)
In this model, a toothed metal disc of radius 30 mm links to a smaller toothed metal disc of radius 12mm. Rotating a crank beneath the baseplate rolls the smaller disc around the outer edge of the larger disk. The model illustrates three curves that may be generated by the motion of a point at a fixed distance along the radius of a circle when the circle rolls around the outer edge of a larger circle.
The green point within the smaller circle (at radius 4mm) produces the green curve on the glass overlay of the model. The blue point on the circumference of the smaller circle (in this special case, the curve is known as an epicycloid) produces the blue curve. The third, represented by a red curve on the glass, is on the extension of a radius of the smaller circle (20mm). As the smaller circle rolls, the point moves inside the larger circle. The German title of this model is: Erzeugung der Epitrochoiden als solche mit freiem Centrum.
Schilling, Martin, Catalog Mathematischer Modelle für den höheren mathatischen Unterricht, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 1.
Chironis, Nicholas, Gear Design and Applications, 1967, p. 160.
Davis, W.O., Gears for Small Machines, 1953, p. 9.
Grant, George, Teeth of Gears, 1891, p. 69.
Material for educators can be found online at the Durango Bill website.
Online demonstrations can be found at the Wolfram website Mathworld.
Currently not on view
date made
ca 1900
Schilling, Martin
place made
Physical Description
metal (overall material)
glass (overall material)
paper (overall material)
leather (overall material)
overall: 5.8 cm x 22 cm x 27 cm; 2 9/32 in x 8 21/32 in x 10 5/8 in
ID Number
catalog number
accession number
Credit Line
Gift of the Department of Mathematics, The University of Michigan
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Kinematic Models
Data Source
National Museum of American History, Kenneth E. Behring Center


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