Peaucellier Inversor, Kinematic Model by Martin Schilling, series 24, model 10, number 349

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the tenth 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. 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.

In 1864 French engineer Charles Nicolas Peaucellier (1832-1913) created a seven-bar linkage which succeeded in producing pure linear motion. Since then, such seven-bar linkages are often referred to as “Peaucellier cells” or a “Peaucellier’s inversor.” His discovery was the first solution of what was referred to as the problem of parallel motion: converting rotational to linear motion using only “rods, joints and pins.”

This linkage is constructed of seven metal armatures (two longer arms of 15 cm, four shorter arms of 5 cm) hinged so as to create a kite shape with a rhombus (diamond) shape at the top of the kite. This horizontal assembly is then attached to a central vertical axis that is rotated by turning a crank below the baseplate. Fingerholds are attached to the two primary hinge points to allow the linkage to be articulated from above. Below these fingerholds are metal points used to trace the motion of the linkage on the paper.

The bottom (tail end) of the kite is fixed on a circle. As the dial is turned or the fingerholds are moved, the point internal to the linkage traces the circle, resulting in the point at the top of the kite tracing a straight line. The seven-bar linkage works by keeping one end of the linkage (the tail end of the kite) fixed on a circle. As the center point traces around the circle, the point at the opposite end of the linkage (the top of the kite) traces a straight line. The German title for this model is: Inversor von Peaucellier 1864.

This model is marked Halle a.S., but this is lined through and has a Leipzig stamp. In or after 1903 Schilling moved the business from Halle to Leipzig. On top of the mounting plate is an aged paper sheet showing the name and maker of the linkage. Printed on the paper are a black circle and a red circle to show the circular path; a red curve; and, what was most likely a black linear path, but which is now worn through the paper with use.

References:

Johnson, Wm. Woolsey, “The Peauceller Machine and Other Linkages,” The Analyst, Vol 2, No. 2, Mar 1875, pp. 41-45.

Roberts, David Lindsay, “Linkages, A Peculiar Fascination” in Tools of American Mathematics Teaching, 1800-2000, Kidwell, P. A. et.al., Johns Hopkins University Press, 2008, pp. 233-242.

Schilling, Martin, Catalog Mathematischer Modelle für den höheren mathatischen Unterricht, Halle a.S., Germany, 1911, pp. 56-57. Series 24, group IV, model 10.

Mathematics and interactive Java applet can be found at http://www.cut-the-knot.org/pythagoras/invert.html\

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