Neusis construction

The neusis is a geometric construction method that was used in antiquity by Greek mathematicians.

The neusis construction (from Greek νεῦσις from νεύειν neuein "incline towards"; plural: νεύσεις neuseis) consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P.

A neusis construction might be performed by means of a 'neusis ruler': a marked ruler that is rotatable around the point P (this may be done by putting a pin into the point P and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue eye) indicates the distance a from the origin. The yellow eye is moved along line l, until the blue eye coincides with line m. The position of the line element thus found is shown in the figure as a dark blue bar.

Point P is called the pole of the neusis, line l the directrix, or guiding line, and line m the catch line. Length a is called the diastema (διάστημα; Greek for "distance").

Neuseis have been important because they sometimes provide a means to solve geometric problems that are not solvable by means of compass and straightedge alone. Examples are the trisection of any angle in three equal parts, the doubling of the cube, and the construction of a regular heptagon, nonagon, or tridecagon (polygons with 7, 9, or 13 sides). Mathematicians such as Archimedes of Syracuse (287–212 BC) and Pappus of Alexandria (290-350 AD) freely used neuseis; Sir Isaac Newton (1642-1726) followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use.

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