The neusis construction consists of fitting a lineelement of given length in between two given lines, 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. 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 indicates the distancea 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.
Use of the neusis
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. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis; Sir Isaac Newton followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use. A regular n-gon is known to be constructible with the neusis for n = while a regular n-gon is known to be non-constructible with the neusis for n = with the status still an open question for n =
Waning popularity
, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. The principle to avoid neuseis whenever possible may have been spread by Hippocrates of Chios, who originated from the same island as Oenopides, and who was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him Euclid too shunned neuseis in his very influential textbook, The Elements. The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were:
constructions that in addition to this use conic sections ;
constructions that needed yet other means of construction, for example neuseis.
In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution. Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as "a not inconsiderable error".
