Total curvatures of a closed curve in Euclidean $n$-space

A classical result by J. W. Milnor states that the total curvature of a closed curve $C$ in the Euclidean $n$-space is the limit of the total curvatures of polygons inscribed in $C$. In the present paper a similar geometric interpretation is given for all total curvatures $\int_{C}\vert\kappa _{r}\vert\mathrm{d}s$, $r=1,\ldots,n-1$.