Nicomedes's Geometric Construction of Principal Cube Root

This Demonstration shows Nicomedes's method for constructing a cube root using a marked ruler. (Nicomedes was born about 270 BC.) This is actually the real principal cube root. Every nonzero number also has a pair of complex conjugate cube roots.

Let have sides of length , 1 and 1, where with . Let be the midpoint of and let be the midpoint of . Let be a point on the line through parallel to . Let be a point on the ray and be the intersection of segments and . If , then .

This Demonstration is based on [1, pp. 128, 129]. The proof from [1] follows.

Let the line parallel to that passes through intersect at . So . Then, since bisects , . Also, since , we have . With , by two applications of the Pythagorean theorem,

,

which reduces to the quartic equation

.

Fortunately, this quartic easily factors as . Since , .

Reference

[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.