Sylvester-Kempe Inversor, Kinematic Model by Martin Schilling, series 24, model 12, number 351

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

Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight-line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context, “Inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor” is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

A generalization of Hart’s Inversor, the Sylvester-Kempe Inversor is also known as a Quadruplane inversor and creates linear motion from circular motion. English mathematicians James Sylvester (1814-1897) and Alfred Kempe (1849-1922) developed the geometric theory behind these linkages in the 1870s. Kempe proved that every algebraic curve can be generated by a linkage using a Watt’s curve, after Scottish engineer James Watt (1736-1819).

Unlike the other Schilling linkages in the collection, this one is not made of armatures. It consists of linked triangular metal plates (two large and two small). The smallest triangle is attached to the baseplate at a stationary pivot point. The triangles are linked together at the vertices to form a chain of triangles (small-large-small-large). As with the other linkages, this model has an armature that is attached to a small hand crank on the underside of the baseplate and attached to the vertex of one of the larger triangles that allows the linkage to rotate. It can also be moved by using one of two fingerholds attached to the top of two of the triangles at a vertex.

As the linkage is rotated, a pin where the armature attaches to the large triangle traces out a circle, visible in the image. At the same time, a pin under the fingerhold on the opposite large triangle traces a straight line from left to right across the baseplate, also seen in the image. The German title of this model is: Inversor von Sylvester und Kempe. The nameplate on the model gives a date of 1875 for this model, most likely the date of discovery by Sylvester and Kempe.

Reference:

Schilling, Martin, Catalog Mathematischer Modelle für den höheren mathatischen Unterricht, Halle a.s., Germany, 1911, pp. 56-57. Series 24, group IV, model 12.

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