We report accurate Ab Initio studies of the static dipole polarizabilities of anthracene and phenanthrene. Geometries were optimized at the HF/6-311G(3d,2p) level of theory. Dipole polarizabilities were calculated at the HF/6-311++G(3d,2p) and BLYP/6-311++G(3d,2p) levels of theory, using HF/6-311G(3d,2p) geometries. The calculated dipole polarizabilities for anthracene are compared with experiment. Inclusion of electron correlation using the BLYP procedure increases the L and M components of the dipole polarizability, but not the perpendicular (N) component. Examination of corresponding BLYP results for the polyacene series benzene, naphthalene and anthracene shows that the normal component of the dipole polarizability scales linearly with the number of benzene ring units, with an increment of 20.8 au. The medium component also scales linearly with an increment of 42.8 atomic units. The long component does not scale linearly. Semi-emiprical AM1 calculations are also given for comparison; the normal component of the dipole polarizability tensor is poorly represented by such calculations.