Geometric construction
The neusis construction consists of fitting a line element of given length () in between two given lines ( and ), in such a way that the line element, or its extension, passes through a given point . That is, one end of the line element has to lie on , the other end on , while the line element is "inclined" towards . Point is called the pole of the neusis, line the directrix, or guiding line, and line the catch line. Length is called the ''diastema'' (). A neusis construction might be performed by means of a marked ruler that is rotatable around the point (this may be done by putting a pin into the point 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 from the origin. The yellow eye is moved along line , until the blue eye coincides with line . The position of the line element thus found is shown in the figure as a dark blue bar. If we require that both two marking of the ruler must land on straight line, then the construction is called line-line neusis. Line-circle neusis and circle-cirle neusis are defined in similar way. The line-line neusis gives us precisely the power to solve quadratic and cubic (and hence also quartic) equations while line-circle neusis and circle-circle neuis is strictly more powerful than line-line neusis. Technically, any point generated by either the line-circle neusis or the circle-circle neusis lies in an extension field of the rationals that can be reached by a tower of fields in which each adjacent pair has index either 2, 3, 5, or 6 while the adjacent-pair indices over the tower of the extension field of line-line neusis are either 2 or 3.Trisection of an angle
Let ''l'' be the horizontal line in the adjacent diagram. Angle ''a'' (left of point ''B'') is the subject of trisection. First, a point ''A'' is drawn at an angle's ray, one unit apart from ''B''. A circle of radius ''AB'' is drawn. Then, the markedness of the ruler comes into play: one mark of the ruler is placed at ''A'' and the other at ''B''. While keeping the ruler (but not the mark) touching ''A'', the ruler is slid and rotated until one mark is on the circle and the other is on the line ''l''. The mark on the circle is labeled ''C'' and the mark on the line is labeled ''D''. Angle ''b'' = ''CDB'' is equal to one-third of angle ''a''.Use of the neusis
''Neuseis'' have been important because they sometimes provide a means to solve geometric problems that are not solvable by means ofRegular polygons
In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower of fields over , , such that the degree of the extension at each step is no higher than 6. Of all prime-power polygons below the 128-gon, this is enough to show that the regular 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, 89-, 103-, 107-, 113-, 121-, and 127-gons cannot be constructed with neusis. (If a regular ''p''-gon is constructible, then is constructible, and in these cases ''p'' − 1 has a prime factor higher than 5.) The 3-, 4-, 5-, 6-, 8-, 10-, 12-, 15-, 16-, 17-, 20-, 24-, 30-, 32-, 34-, 40-, 48-, 51-, 60-, 64-, 68-, 80-, 85-, 96-, 102-, 120-, and 128-gons can be constructed with only a straightedge and compass, and the 7-, 9-, 13-, 14-, 18-, 19-, 21-, 26-, 27-, 28-, 35-, 36-, 37-, 38-, 39-, 42-, 52-, 54-, 56-, 57-, 63-, 65-, 70-, 72-, 73-, 74-, 76-, 78-, 81-, 84-, 91-, 95-, 97-, 104-, 105-, 108-, 109-, 111-, 112-, 114-, 117-, 119-, and 126-gons with angle trisection. However, it is not known in general if all quintics (fifth-order polynomials) have neusis-constructible roots, which is relevant for the 11-, 25-, 31-, 41-, 61-, 101-, and 125-gons.Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164, Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible; the 25-, 31-, 41-, 61-, 101-, and 125-gons remain open problems. More generally, the constructibility of all powers of 5 greater than 5 itself by marked ruler and compass is an open problem, along with all primes greater than 11 of the form ''p'' = 2''r''3''s''5''t'' + 1 where ''t'' > 0 (all prime numbers that are greater than 11 and equal to one more than a regular number that is divisible by 10).Waning popularity
T. L. Heath, 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 himSee also
* Alhazen's problem *References
*R. Boeker, 'Neusis', in: ''Paulys Realencyclopädie der Classischen Altertumswissenschaft'', G. Wissowa red. (1894–), Supplement 9 (1962) 415–461.–In German. The most comprehensive survey; however, the author sometimes has rather curious opinions. * T. L. Heath, ''A history of Greek Mathematics'' (2 volumes; Oxford 1921). * H. G. Zeuthen, ''Die Lehre von den Kegelschnitten im Altertum'' The Theory of Conic Sections in Antiquity(Copenhagen 1886; reprinted Hildesheim 1966).External links