Gravitational fields and the cosmological constant in multidimensional Newtonian universes

The distance dependence of gravity is found in Newtonian universes with any number n of space dimensions. Two independent derivations given are based either on (i) requiring that a (hyper) spherical mass gravitate as if all its mass were concentrated at its center, or (ii) using the field equations of general relativity with the cosmological constant Λ. Both approaches lead to identical results. The gravity field at distance r from a point mass has two parts, one going as r1-n, the other as r, i.e., Hooke's law. The Hookian field obeys a novel form of Gauss's (flux) law, and is closely related to Λ. The simple mechanical interpretation which emerges gives insight into the meaning of Λ and helps counteract certain prevalent misconceptions.