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Diffraction of a wave by an aperture of any shape in a thin screen is treated by a new method—``the geometrical theory of diffraction.'' This is an extension of geometrical optics which accounts for diffraction by introducing new rays called diffracted rays. They are produced when incident rays hit the aperture edge and they satisfy the ``law of diffraction.'' A field is associated with each ray in a quantitative way, by means of the optical principles of phase variation and energy conservation. In addition ``diffraction coefficients'' are introduced to relate the field on a diffracted ray to that on the corresponding incident ray. By this method a simple formula is obtained for the field diffracted by any aperture. It yields the field in the aperture, the diffraction pattern and the transmission cross section. Explicit formulas and numerical results are given for slits and circular apertures. The accuracy of the results increases as the wavelength decreases, but they are useful for wavelengths even as large as the aperture dimensions.