A metric d is said to have maximal symmetry iff its isometry group is not properly contained in the isometry group of any metric equivalent to d. Theorem. Every convex, two-point homogeneous metric for which small spheres are connected has maximal symmetry. Corollary. The standard metrix on \({\mathbb{E}}^ n\), \({\mathbb{S}}^ n\), \({\mathbb{R}}{\mathbb{P}}^ n\), \({\mathbb{C}}{\mathbb{P}}^ n\), \({\mathbb{Q}}{\mathbb{P}}^ n\), \({\mathbb{H}}^ n\), \({\mathbb{C}}{\mathbb{H}}^ n\), \({\mathbb{Q}}{\mathbb{H}}^ n\), Cay \({\mathbb{P}}^ 2\), Cay \({\mathbb{H}}^ 2\) have maximal symmetry.