GR can be interpreted as a theory of evolving 3-geometries. A recent such formulation, the 3-space approach of Barbour, Foster and Ó Murchadha, also permits the construction of a limited number of other theories of evolving 3-geometries, including conformal gravity and strong gravity. In this paper, we use the 3-space approach to construct a 1-parameter family of theories which generalize stron...