Measurements are presented of the total neutron cross sections of naturally occurring ₅₇La,₅₈Ce, ₅₉Pr, ₆₀Nd, ₆₂Sm, ₆₄Gd, ₆₆Dy, ₆₇Ho, ₆₈Er and ₇₀Yb over the energy range of 0.7 MeV to 9.0 MeV. The results are compared with theoretical cross sections evaluated using the spherical generalised optical potential of Wilmore and Hodgson. Considerable resonant structure is found in the total cross section of ₅₈Ce and there is evidence of an abrupt onset of nuclear deformation as the neutron number reaches 90.