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**Extra resources for Problems: 144 problems of the Austrian-Polish Mathematics Competition, 1978-1993**

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Even now one scarcely thinks of the equality of angles when reflection results in color, as in looking at the blue sky or a red rug. The law of reflection is suggested by a study of images rather than colors. " The quantitative principle in such cases therefore is not explicitly stated; but considerations in analogous acoustic and mechanical situations make it clear that the optical law was indeed known. Aristotle on several occasions pointed out analogies between the phenomena of sight and sound, 23 and hence the following passage on echoes may, by implication, be regarded as directly relevant: Why is it that the voice, which is air that has taken a certain form and is carried along, often loses its form by dissolution, but an echo, which is caused by such air striking on something hard, does not become dissolved, but we hear it distinctly?

But how short-lived was the temporal power of the latter in comparison with the cultural hegemony of the former! Within a few years of his death the empire of Alexander crumbled. The territory was divided among his generals and, disorganized and distrustful of each other, the Greek states succumbed to the relentless expansion of Rome. c. Corinth, as well as Carthage, was utterly destroyed, and Rome became master of western civiliza. tion. " Politically the Mediterranean had become Roman; intellectually it remained Hellenic.

Aristotle has recourse to an entirely different line of reasoning, one which dispenses with the meteorological sphere and substitutes for it a vertical plane containing the bow. Let B, G, and R be the highest points, respectively, of the blue, green, and red bands in the secondary bow, arranged on a vertical line; and letS be the sun and 0 the observer (Fig. 9). 48 Thus the difficulty presented by the inverted ordering of the colors is resolved; but Aristotle, FIG. -An Aristotelian explanation for the ordering of the colors in the secondary rainbow.