A Brief History of Impossibility

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Sooner or later every student of geometry learns of three impossible problems: 1. Trisecting the angle: Given an arbitrary angle, construct an angle exactly one-third as great. 2. Duplicating the cube: Given a cube of arbitrary volume, find a cube with exactly twice the volume. 3. Squaring the circle: Given an arbitrary circle, find a square with the same area. These problems originated around 430 BC at a time when Greek geometry was advanc ing rapidly. We might add a fourth problem: inscribing a regular heptagon in a circle. Within two centuries, all these problems had been solved (see [3, Vol. I, p. 218-270] and [1] for some of these solutions). So if these problems were all solved, why are they said to be impossible? The im possibility stems from a restriction, allegedly imposed by Plato (427-347 BC), that geometers use no instruments besides the compass and straightedge. This restriction requires further explanation. For that, we turn to Euclid (fl. 300 BC), who collected and systematized much of the plane geometry of the Greeks in his Elements. Euclid's goal was to develop geometry in a deductive manner from as few basic assumptions as possible. The first three postulates in the Elements are (in modernized form): 1. Between any two points, there exists a unique straight line. 2. A straight line may be extended indefinitely. 3. Given any point and any length, a circle may be constructed centered at the point with radius equal to the given length. These three postulates correspond to the allowable uses of compass and straightedge: to draw a line that passes through two given points; to extend a given line segment indefinitely; and to draw a circle about any given point with any given radius. To solve a problem using compass and straightedge means to use only these operations, repeated a finite number of times. The construction's validity can then be proven using only the postulates of Euclidean geometry.