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Old Euclid drew a circle On a sand-beach long ago. He bounded and enclosed it With angles thus and so. His set of solemn greybeards Nodded and argued much Of arc and circumference, Diameter and such. A silent child stood by them From morning until noon Because they drew such charming Round pictures of the moon.
Vachel Lindsay, EuclidMany mathematical curves have intriguing properties. In visiting the world of curves, one enjoys three complementary views: some curves arise as geometrical shapes appearing in nature, others come from the observation of dynamic phenomena, and a wide range of curves result from mathematical ingenuity [Wells, 1991].
Our aim in this chapter is to present a selection of attractive proofs related to various extraordinary properties of some curves. We begin with some theorems about lunes, a shape that appears in nature as phases of the moon.
Squarable lunes
As we noted in the previous chapter, the ancient Greeks were concerned with the notion of quadrature, constructing with straightedge and compass a square equal in area to a given figure. This process is also called squaring the figure, and a figure that can be squared is called squarable.
A lune (from the French word for moon) is a concave region in the plane bounded by two circular arcs. A convex region bounded by two circular arcs is a lens. See Figure 9.1 for an illustration of two lunes (gray) and a lens (white).
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