R 3 be any 2-plane, which is not a subspace. We prove that if K ⊂ R 3 is a Borel set with dimH K ≤ 3 2 , then dimH πe(K) = dimH K for H 1 almost every e ∈ S 2 ∩W , where H 1 denotes the 1-dimensional Hausdorff measure and dimH the Hausdorff dimension. This was known earlier, due to Järvenpää, Järvenpää, Ledrappier and Leikas, for Borel sets K with dimH K ≤ 1. We also prove a partial result for sets with dimension exceeding 3/2, improving earlier bounds by D. Oberlin and R. Oberlin. Recent evidence suggests that, in (MM1)-(MM2), the 2-dimensional measure H 2 on S 2 can be replaced by length measure on certain curves Γ ⊂ S 2. The main result in [5] proved this for part (MM1), whenever Γ is a circle, but not a great circle (the great circles are a "degenerate" case, having non-trivial orthogonal complement). We refer the reader to [5] for a broader introduction, and earlier results, on the projections ρ e. In this paper, we consider part (MM2) in the same setting: Theorem 1.1. Let W ⊂ R 3 be a 2-plane, which is not a subspace. If K ⊂ R 3 is a Borel set, then dim H π e (K) ≥ min dim H K, 1 + dim H K 3 for H 1 almost every e on the circle S W = S 2 ∩ W. In particular, the projections π e , e ∈ S W , preserve H 1 almost surely the dimension of at most 3 2-dimensional Borel sets.