An Investigation of Historical Geometric Constructions

Editor’s note: This article was published in July 2006.

From ancient Greek times, mathematicians have considered three famous geometric construction problems. These problems are: (1) the duplication of the cube – construct the edge of a cube whose volume is twice that of a given cube; (2) the trisection of any angle – construct an angle that equals one-third of a given angle; and (3) the squaring of a circle – construct a square whose area equals the area of a given circle. Given the tools at the time, a compass and an unmarked ruler, these problems challenged mathematicians for centuries.

The best Greek intellect was bent upon them; Arabic learning was applied to them; some of the best mathematicians of the Western Renaissance wrestled with them.  Trained minds and untrained minds, wise men and cranks, all endeavoured to conquer these problems which the best brains of preceding ages had tried but failed to solve. (Cajori, 1924, p. 54)

Although these three problems are closely linked, we choose to examine the latter two problems. We must divulge that the given problems do not have solutions if we limit ourselves to constructions using only compass and straightedge. However, using curves created by other means the Greeks resolved all three problems.