The Three Construction Problems of Antiquity

There are three geometric construction problems originally proposed by the ancient Greeks. These problems have become famous because the Greeks were unable to solve them and the verdict on whether they could be solved was not handed down until relatively recently. These three problems are referred to as "duplicating the cube", "trisecting an angle", and "squaring the circle". They consist of performing the following constructions by using only a straightedge and compass:

Duplicating the Cube

Construct a cube having twice the volume of a given cube. This involves constructing the line the cube root of 2 times the length of a given line segment.

Trisecting an Angle

Devise a method by which any angle may be trisected.

Squaring the Circle

Construct a square equal in area to a given circle (or the reverse).

It was only shown in the 19^th century that these problems are impossible, and it was by algebraic methods. It can be shown that the only lengths that can be constructed by means of compass and straightedge are those that are successions of square roots applied to rational numbers. These numbers correspond to algebraic equations of even degree. The cube root of 2, however, is the root of a third-degree equation, and cannot be the solution of an equation of even degree. Similarly, trisecting an angle involves finding a solution to cos 3θ = 4cos³θ - 3cosθ which boils down to another third-degree equation.

It is harder to prove that the third problem cannot be solved. The solution involved showing that π is a transcendental number. Since a transcendental number cannot be the solution of any algebraic equation, this construction too is impossible.