  Aristotle treats the science at the lowest level, descriptions of the rainbow, astronomical phenomena, and acoustical harmonics, as descriptive, providing the fact that something is the case, but not the explanation, which is provided by the higher science. It is very easy to speculate how Aristotle would fill in the relations in the table; e.g., would he put stereometry below geometry, as Plato does in Rep. vii? Similarly, the explanatory relation between mathematical optics and geometry is not the same as the relation of optics to empirical optics. This example of the rainbow seems to refer to the argument in Meteorology iii.5, where the observed fact that rainbows are never more than a semicircle (true in flat lands) is explained by a proof in optics that is thoroughly geometrical in character. Once the basic set-up and principle of reflection is provided, the rest is geometrical.