Ancient geometers geometrically constructed the number $\pi$ using a special curve, called the quadratrix of Hippias (or Dinostratus). One way to construct the quadratrix is by tracing the path of the intersection point between a radius uniformly rotating and a segment that moves uniformly along the vertical. Both motions must be synchronized so that they occur at the same time. The construction of this curve by motions requires two degrees of freedom. Is there a way to construct the quadratrix using one motion only, i.e. by a one-degree-of-freedom construction?
The construction of the quadratix using the quadratix compass (as described in Wikipedia page) has two degrees of freedom.
The description of the quadratrix by composition of motions can be found in Pappus. The quadratrix is a transcendental curve with polar equation $r(\phi) = \frac{2a\phi}{\sin\phi}$ (by simplicity, we can consider the quadratrix in a circle of radius $a=1$). As a result, it can be seen that the ordinates of each point on the quadratrix are proportional to the angles made by the corresponding radii.
According to Kempe's theorem, the quadratrix cannot be generated by any articulated system of rulers, or linkages which impart one degree-of-freedom motions. However, one could use a different mechanical device to construct this curve by one motion, for example a device that employs strings or threads. A theoretical machine to construct the quadratrix via strings can be described as follows.
Let a circular disk of center $A$ and radius $r= 2/\pi$ be given in the plane. Let us then mark a point $B$ on the circumference and insert a ruler $AC$ pivoting around the center $A$. A second ruler is to be placed, outside the circle and parallel to $AB$, in such a way that it can only move upwards and downwards (For simplicty's sake, in the figure we do not represent the mechanism that ensures the parallelism). Starting from point $C$ on the circumference, let us stretch an inexstensible string around the circumference of the circle, and fasten its free end to $D$ on the ruler $d$, in order to keep the string in tension. The intersection of $d$ with the extension of the radius $AC$ will determine a point $E$. If we move the ruler $d$, $BD$ will rectify the arc $BC$ and the point $E$ will describe the quadratrix of a circle with radius $r=1$. In fact, if we call $\alpha$ the angle $\angle BAC$, then $BD = 2 \alpha/\pi$. The variation of the angle is proportional to the distance between the line $d$ and $A$, and there are suitable boundary conditions (when $\alpha = 0$, i.e. $C = B$, the line d passes through $A$; when $\alpha = \pi/2$, the distance between $d$ and $A$ is $1$). Hence the curve traced by $E$ is a quadratrix. However, this construction involves a circularity, for the reason that the realization of the device here described depends on the construction of a circle with radius $2/\pi$, and therefore on the rectification of the circumference. But this is exactly the problem that the quadratrix was invented for.