Let D be a compact set in the plane, t a real number, and Dt the linear set {x + ty|x + iy ε D}. We are interested in the Hausdorff dimensions of D and Dt, and assume that dim D = d ≤ 1. A number t is “exceptional” if dim Dt < d; the exceptional numbers form a Borel set of d-dimensional measure zero [3]. (Marstrand [4, p. 268] proves a similar conclusion for the Lebesgue measure of the exceptional set.) In this note we exhibit a planar set D of Hausdorff dimension d and a linear set E for which dim Dt ≤ r < d for every t in E, but dim E > 0. The method does not come very close to the probable truth, that the set E can have dimension r. But perhaps the result can be improved by more subtle calculation, for example, as in Jarník [2].