Given the ease with which we can geometrically multiply and divide straight-line segments, i would like to ask for guidance on the same problems on the circle, not necessarily using "straight edge and compass", but any possible method:
Given two arcs of length x and y on the unit circle, construct arcs of length $x\cdot y, \frac{x}{y}, \,\,\frac{x}{n} \text{ for }n=2,3,4,\ldots$.
The only results i'm aware of are those of the theory of constructible polygons, with straight edge and compass.
On
OK. Let's assume a unit circle. Given points $X$ and $Y$ with arclengths $x\in \Bbb R$ and $y\in \Bbb R$ measured counterclockwise from the initial point $P = (1, 0)$, compute, using the axioms of the real numbers, the number $xy$. Then let $U = (\cos(xy), \sin(xy))$; the arclength counterclockwise from $P$ to $U$ is then $xy + 2k \pi$, for some integer $k$. The same approach works for all the other possibilities.
Use a thick disk and a thread. You can roll/unroll and straighten the thread to convert from arc to line segment.
Alternatively, use an Archimedes' spiral. The polar angle and the modulus perform the same transformation.