An elementary discussion of the Abbe theory of microscopic vision is given, suitable for teaching purposes. The effect of the phases in the back focal plane of the objective is briefly outlined. The consequence of limiting the aperture of the system in this plane is shown to be an image marred by diffraction.The relationship to Fourier's theorem having been indicated, the Gibbs phenomenon is next described. A standard formula for the finite sum of a Fourier series is taken and adapted to the case of imaging a suitable grating with a microscope objective limited by a slit aperture. If the grating slits are very narrow compared with the opaque region separating them, the image of each is shown to be marred by a set of fringes, due to the Gibbs phenomenon, exactly equivalent to the diffraction fringes from the single slit in the microscope objective.