The study of irrotational incompressible flows about thin geometries can be carried out using the well known perturbation procedures. In two-dimensional flows exact solutions based on mappings can be used to compare the accuracy of first order solutions. For most airfoil sections a first order perturbation solution is not sufficiently accurate in representing the pressure and velocity distribution, especially about the leading edge. For three-dimensional flows exact solutions are rare and for more complex problems such as ship wave resistance formulations an exact solution does not exist for comparison of results. In this last case second-order solutions exist but are very difficult to calculate. Therefore, it would appear advantageous to improve first-order calculations. To this end a perturbation method that incorporates the geometric properties of the body is studied. This method is applied to a symmetric Joukowski airfoil and to an elipse. This method, here called the “geometrically-consistent linearization method” predicts the leading edge pressure variations correctly in the two cases studied and appears to be superior to the classical first order solutions. An iterative solution following this procedure further improves the calculation especially for thicker foils. The method discussed and the following iteration procedure seem to form an efficient numerical solution to airfoil flow problems.