**Wittgenstein's rod** is a geometry problem discussed by 20th-century philosopher Ludwig Wittgenstein.

## Description

A ray is drawn with its origin 'A' on a circle, through an external point *S* and a point *B* is chosen at some constant distance from the starting end of the ray; what figure does *B* describe when all the initial points on the circle are considered? The answer depends on three parameters: the radius of the circle, the distance from the center to *S* and the length of the segment *AB*. The shape described by *B* can be seen as a 'figure of eight' which in some cases degenerates to a single lobe looking like an inverted cardioid.

If *B* remains on the same side of *S* with respect to the center of the circle, instead of a ray one can consider just a segment or the rod 'AB'.

Wittgenstein sketched a mechanism and wrote:

While the point A describes a circle, B describes a figure eight. Now we write this down as a proposition of kinematics.

When I work the mechanism its movement proves the proposition to me; as would a construction on paper.

The proposition corresponds e.g. to a picture of the mechanism with the paths of the points A and B drawn in. Thus it is in a certain respect a picture of that movement. It holds fast what the proof shows me. Or - what it persuades me of.

This text has been included among the notes selected for publication in *Remarks on the Foundations of Mathematics* and the editors have dated in the as spring of 1944.^{[1]}

## Related mechanism

**Wittgenstein's rod** is a generalization of Hoeckens linkage.

## Animations

## References

**^**Wittgenstein L.,*Remarks on the Foundations of Mathematics*, edited by G.H. von Wright and Rush Rhees, Oxford: Blackwell 1998, ISBN 0-631-12505-1, sect V, §72, p.434