For two-dimensional Eden clusters grown on flat substrates the way the perimeter per unit length and the surface width depend on the orientation of the substrate with respect to the lattice is measured. The evolution of the perimeter per unit length during growth is shown to be independent of the substrate length. For the surface width strong corrections to scaling are observed. The anisotropy of three-dimensional Eden clusters and its implications for the scaling of the surface width are investigated. The largest two and three-dimensional clusters studied are more than one order of magnitude larger than in previous simulations.