The local nonsimilarity solution method has been applied to solve for natural convection on a vertical cylinder for conditions where there are large deviations from the flat plate results. To assess and insure the accuracy of the results, solutions were obtained for three levels of truncation of the governing equations. The solutions were carried out for Pr = 0.7.33 and for a range of cases extending from small deviations from the flat plate to a factor-of-four deviation between the local heat fluxes for the cylinder and the flat plate. The results provided by the local similarity solutions were found to be highly accurate over this range. Comparisons were made with available results for the local and surface-integrated heat transfer, and solution methods were identified which appear promising for application beyond the range studied here. A presentation of representative temperature and velocity profiles showed only small deviations between the local similarity solutions and those for the third level of truncation.