The Duplicators, Part I: Eutocius's Collection of Cube Duplications

Overview

​It is well known that the three classic geometry problems—duplicating the cube, trisecting the angle, and squaring the circle—are impossible with the classical tools of straightedge and compass only. However, two things are less well known: first, that the first two problems are solvable if additional methods or tools are permitted; second, that there are a wealth of these solutions from the ancient and medieval world. This paper focuses on the solutions by Greek mathematicians recorded in Eutocius of Ascalon’s commentary on Archimedes’ work On the Sphere and the Cylinder. Many of the techniques employed will likely be foreign to most readers, particularly the neusis and neusis-like constructions. The ones that use conic sections will be more familiar. The reader may also find the use of mechanical instruments (or abstract conceptualizations of them) unusual but interesting. The majority of the paper is comprised of my translation of Eutocius, along with copious notes and interactive GeoGebra diagrams.