In this article I consider the optimum performance at the maximum power output of internally and externally irreversible heat engines of finite size in which heat transfer is proportional to the difference of temperature to an n-power. For the case n = I, where heat transfer is Newtonian heat conduction, the hot-end / cold-end heat-exchanger size ratio is less than unity. For the case n = 4, which corresponds to radiative heat transfer the values of design parameters are obtained from the numerical solution of two coupled nonlinear equations. For a radiative heat engine of finite size, the upper limit to the optimum efficiency at maximum power output is found to be 0.339.