What is a simple, elementary proof of the following result?

*A continuously differentiable map from the unit sphere $S^n \subset \mathbb{R}^{n+1}$ $(n > 1)$ to itself that preserves volumes and sends great circles to great circles is an isometry.*

I have a simple, but non-elementary proof of a more general result:

*A continuously differentiable map from a Zoll Riemannian or reversible Finsler manifold to itself that preserves volumes and sends geodesics to geodesics is an isometry.*

I don't think the generality makes the result more interesting (non-isometries sending geodesics to geodesics are not very many except in the case of real projective spaces and spheres) and I wonder if there is some easy, quick argument that takes care of the case of spheres with their canonical metrics.