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By means of noise reduction we investigate the scaling behaviour of the surface width w of d-dimensional Eden clusters growing on flat substrates of linear extent L. We find that w scales for small average deposit height h as hβ, and in the steady state as Lα. Our results for d = 3 (α = 0.33 ± 0.01, β = 0.22 ± 0.02) and for d = 4 (α = 0.24 ± 0.02, β = 0.146 ± 0.015) support the scaling relation α(1 + 1/β) = 2. They suggest that α instead of being superuniversal might be equal to 1/d.