# The Duplicators, Part I: Eutocius' Collection of Cube Duplications - On the Methods of Solution

Author(s):
Colin B. P. McKinney (Wabash College)

The methods of solution presented here are likely to seem strange to a modern reader. We have several involving motion, two involving the use of conic sections (which themselves ultimately derive from motion, since they derive from a cone), and several that involve motion in a way that has a mechanical feeling, suggesting their adaptability to practical computation. A reader who has previously studied Greek mathematics may have come across the term neusis to describe a particular type of solution to various problems. In the usual sense, neusis involves finding a line from a given point, so that the segment cut off by two given lines is equal to a given segment. The two lines could be either straight or curved.

For example, in the following figure, we have two curved lines, and a given point A. We seek the particular configuration where the segment between the two curves is equal to some given length. There is a bit of a Goldilocks feel here: some segments, like BC, are too large; others, like DE, are too small; but one, say FG, is just right.

Above: The general setup of a neusis construction.
The point G is movable.

The reader will note however that the methods below do not precisely fit this mold. For this reason, the term neusis-like (coined by Fried and Unguru, in [2001]), seems an apt choice. Nicomedes’ solution, for example, is built around a curve that is defined in such a way that all the segments between it and a straight line are equal to some given segment. Hence using the curve so generated (which Nicomedes calls a conchoid line) in the usual method of neusis would not be particularly illuminating on its own. But the method of generating it was clearly inspired by the method of neusis; thus, Fried and Unguru’s term neusis-like.

Colin B. P. McKinney (Wabash College), "The Duplicators, Part I: Eutocius' Collection of Cube Duplications - On the Methods of Solution," Convergence (April 2016), DOI:10.4169/convergence20160401