Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the eleventh 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.
The Hart’s Inversor, also known as Hart’s Cell or Hart’s Linkage, is a contraparallelogram of four pin-connected links. It is similar to the Peaucellier Inversor, but is a four-bar linkage as opposed to a seven-bar linkage. It was invented and published by Harry Hart (1848-1920) in 1874.
This model is made from four metal armatures, two measuring 9.5 cm, two 16.5 cm, in an “hourglass” configuration (the two longer arms crossing to form the waist of the hourglass) with two congruent triangles meeting at a common vertex.
When the top and bottom arms are parallel to the top and bottom of the baseplate, the triangles are isosceles. The top arm is fixed to the base slightly to the right of its midpoint. Below this fixed point, a fifth arm is attached to a crank below the baseplate and attached to the underside of the upper cross arm slightly above the midpoint. This attachment can be rotated in a circle either by turning the crank or by using the polished fingerhold on the top of the cross arm.
A pin below the fingerhold (now inserted into a piece of cork to avoid tearing the paper covering of the baseplate) traces part of a circle as seen in the image. This causes a fingerhold and pin (also in a piece of cork) on the second cross arm, slightly below its midpoint, to move laterally right and left across the baseplate in a straight-line motion. As the attachment is rotated, the triangles become progressively more scalene.
In addition, this linkage has the following linearity property. When the linkage is in its original (isosceles) configuration, mark four points on each of the four arms such that the four points lie on a vertical line. Fix the top point and allow the second point (below the top point) to trace a circle. This causes the third point to trace a straight line, and all four points will remain colinear regardless of the configuration of the linkage.
The German title of this model is: Inversor von Hart. The name plate on the model gives a date of 1874 for this model, most likely indicating the date of Hart’s discovery.
References:
Schilling, Martin, Catalog Mathematischer Modelle für den höheren mathatischen Unterricht, Halle a.s., Germany, 1911, pp 56-57. Series 24, group IV, model 11.
Online demonstrations for this model can be found at www.cut-the-knot.org
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