Involutes of Circles, Kinematic Model by Martin Schilling, series 24, model 6, number 334


Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the sixth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

An involute of a circle is a curve that is produced by tracing the end of a string that is wrapped around a circle as it is unwound while being kept taut. It is the envelope of all points that are perpendicular to the tangents of a circle.

As with the three trochoidal models, these curves were used in the shaping of gear teeth in the 18th century. Following that, it was discovered that shaping the teeth of gears using the curve formed by the involute of a circle also increases the efficiency of gearage. Surprisingly, there are many applications of noncircular gears, such as elliptical, triangular, and quadrilateral gears. (See model 1982.0795.06.)

In this model a toothed circular gear of radius 13 mm is mounted on the baseplate and can be turned via a crank on the underside of the baseplate. A thick piece of beveled glass is mounted above the apparatus. A dark metal toothed bar 45 mm long is attached to the circular gear so that as the crank turns the circular gear, the toothed bar is forced past the circular gear and rotates round it.

Perpendicular to the bar is a thin clip with three small colored balls. A blue ball is attached at the edge of the bar where the bar will touch the circle and traces the involute of the circle in blue on the glass. A red ball is placed 33mm in front of the toothed side of the bar and produces a “stretched” involute in red. A green ball is 45mm behind the toothed side of the bar traces another “stretched” involute in green. German title is: Erzeugung von Kreisevolventen.


Schilling, Martin, Catalog Mathematischer Modelle für den höheren mathatischen Unterricht, Halle a.s., Germany, 1911, pp 56-57. Series 24, group II, model 6.

Online demo at Mathworld by Wolfram:

Date Made: ca 1900

Maker: Schilling, Martin

Location: Currently not on view

Place Made: Germany

Subject: MathematicsEngineering


See more items in: Medicine and Science: Mathematics, Science & Mathematics, Kinematic Models


Exhibition Location:

Credit Line: Gift of the Department of Mathematics, The University of Michigan

Data Source: National Museum of American History

Id Number: 1982.0795.04Catalog Number: 1982.0795.04Accession Number: 1982.0795

Object Name: geometric model

Physical Description: metal (overall material)glass (overall material)paper (overall material)leather (overall material)Measurements: overall: 5.8 cm x 27 cm x 27 cm; 2 9/32 in x 10 5/8 in x 10 5/8 in


Record Id: nmah_1213883

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