A "perfect" scientific theory may be described as one which proceeds logically from a few simple hypotheses to conclusions which are in complete agreement with observation, to within the limits of accuracy of observation. But the theory is "useful" only in so far as it is possible to obtain conclusions from the hypotheses. As accuracy of observation increases, a theory ceases to be "perfect": modifications are introduced, making the theory more complicated and less "useful". Since we do not willingly surrender the wealth of approximate results furnished by the earlier form of the theory, we find ourselves in the unsatisfactory position of using one theory for one problem and another for another, although the two problems really belong to the same part of science. To rescue ourselves from intellectual confusion, we may admit theories called "ideal", in the sense that they deal with an ideal universe, resembling the actual universe to a fair degree of accuracy and usually corresponding to a limiting case of physical reality.
A critical examination of the history of mathematical physics shows that in truth man has always created "ideal" theories. Nature is much too complicated to be considered otherwise than in a simplified or idealized form, and it is inevitable that this idealization should lead to discrepancies between theoretical prediction and observation. As examples we may mention the mechanical theories of rigid bodies and perfect fluids; neither rigid bodies nor perfect fluids exist in nature. Or we may think of the Newtonian theory of gravitation, long regarded as "perfect", but now "ideal", physically replaced by the "perfect (but not so "useful") general theory of relativity.
Geometrical optics is an ideal theory and a useful one. The discovery that the propagation of light is an electromagnetic phenomenon made the subject of optics coextensive with electromagnetism. We may, however, study certain parts of the subject of optics without reference to electromagnetism, always understanding that there is a limit to the physical accuracy of the results so obtained. It is customary to use the name "physical optics" for the more complex and physically accurate theory, and "geometrical optics " for the simpler ideal theory with which we shall be concerned. It is possible to justify geometrical optics as a limiting case of physical optics, the wave-length of the light in question tending to zero; [M Born, Optik (Berlin, 1933), 45] but we shall be content with the development of geometrical optics on the basis of its own hypotheses, just as it is customary to develop the dynamics of rigid bodies as a separate theory, and not as a limiting case of the dynamics of elastic bodies whose elastic moduli tend to infinity.
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