For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a
canonical parametrization of a path by length. In this paper we provide such a
parametrization for all continuous paths. This parametrization is based on an
alternative notion of path length, which we call $\mathsf{len}$. Like Euclidean
path length, $\mathsf{len}$ is invariant under isometries of $\mathbb{R}^n$, is
monotone with respect to sub-paths, and for any two points in $\mathbb{R}^n$
the straight line segment between them has minimal $\mathsf{len}$ length.
Unlike Euclidean path length, the $\mathsf{len}$ length of any path is
defined (i.e., finite) and $\mathsf{len}$ is continuous relative to the uniform
distance between paths. We use this notion to obtain characterizations of those
families of paths which can be reparameterized to be equicontinuous or compact.
Finally, we use this parametrization to obtain a canonical homeomorphism
between certain families of arcs.
