# Q.E.D.

46 47 PossIble ImPossIbIlItIes doubling, squaring and trisecting Socrates 469 399 B.C.E. once used the first two diagrams below to show how to double a square, and the oracle of Delphi predicted that a plague could be stopped by doubling the cubical altar of Apollo. In the nineteenth century it was proved that doubling a cube as well as the other two notorious geometry problems of squaring a circle and trisecting a general angle are impossible if we require, like the ancient Greeks, that we use only a compass and unmarked ruler. If, however, we are allowed to use other tools, all three problems can be solved. To square a circle, roll it half a revolution on a horisontal opposite, top to construct the long rectangle which has the same area as the circle see page 19. Now, using compass and ruler as indicated, construct the square which has exactly the same area as this rectangle see page 7. Archimedes discovered an ingenious method for trisecting an angle between two intersecting lines opposite, middle using a compass and a ruler with two marks on it Just draw a circle and align the ruler as in the diagram. Then the angle is exactly one third of the angle . Doubling a square amounts to constructing 2 from 1 below, right and doubling a cube involves constructing 32 from 1 lower, opposite. Just trisect and fold the paper unit square, as indicated, to construct 32. Easy to describe but tricky to prove. Can you do it