The three classical Greek problems

The Greeks introduced the systematic foundation of mathematics, and for the first time the study of geometry was not tied to specific points and measurements, such as marking the boundary of a piece of land. They did not use co-ordinate systems, preferring methods obtained by intersecting lines and circles, and hence were interested in 'ruler and compass' constructions.

Given a unit length, a real number $\alpha$ is constructible if a line segment of length $|\alpha|$ can be constructed from the unit length in a finite number of steps using only a ruler (i.e. a straight unmarked edge) and pair of compasses.

• 0 and 1 are clearly constructible.
• A real number $\alpha$ is constructible if and only if $-\alpha$ is constructible.

Problem 1: Is it possible to square the circle?

Explanation of problem: Given a circle of radius 1 unit (and hence of area $\pi$), is it possible to construct a square of the same area and thus to construct the length $\sqrt{\pi}$?

Problem 2: Is it possible to double the cube?

Explanation of problem: Given a cube of edge length 1 unit (and hence volume 1), is it possible to construct a cube of volume twice that of the original cube and thus to construct the length $\sqrt[3]{2}$?

Problem 3: Is it possible to trisect an angle?

Explanation of problem: Is it possible to trisect an arbitrary angle? For it is certainly true that some angles can be trisected! In particular, suppose that we can construct a triangle (so the length of each side is a constructible real number) in which one of the angles in the triangle is $\theta$. Is it always possible to construct a triangle (so each side is again a constructible real number) with one of the angles being $\frac{\theta}{3}$?

The study of roots of polynomials with rational coefficients led to the development of field theory in the 19th century. This finally enabled mathematicians to solve the three classical Greek problems which had been open problems since they were posed in about 450BC. The need to solve equations means that this theory has important applications in many areas of mathematics.

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