  The set of osculating circles of a given curve in $\SS^3$ forms a curve in
the set of oriented circles in $\SS^3$. We show that its
"${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian
structure of the set of circles is proportional to the conformal arc-length of
the original curve, which is a conformally invariant local quantity discovered
in the first half of the last century.
