Ingeometry, theneusis (νεῦσις; from Ancient Greek νεύειν (neuein) 'incline towards'; plural:νεύσεις,neuseis) is ageometric construction method that was used in antiquity byGreek mathematicians.

The neusis construction consists of fitting a straight line element of given length (a) in between two given (not necessarily straight) lines (l andm), in such a way that the extension of the line element passes through a given pointP. That is, one end of the line element has to lie onl and the other end onm while the line element is "inclined" towardsP.

PointP is called the pole of the neusis, linel the directrix, or guiding line, and linem the catch line. Lengtha is called thediastema (Greek:διάστημα,lit. 'distance').

A neusis construction might be performed by means of a marked ruler that is rotatable around the pointP (this may be done by putting a pin into the pointP and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye; this is the origin of the scale division on the ruler. A second marking on the ruler (the blue eye) indicates the distancea from the origin. The yellow eye is moved along linel, until the blue eye coincides with linem.

If we require both linesl andm to be straight lines, then the construction is called line–line neusis. Line–circle neusis and circle–circle neusis are defined analogously. 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 neusis are 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.^[1]

Starting with two lines${\displaystyle {\ell }_1}$ and${\displaystyle {\ell }_2}$ that intersect at angle${\displaystyle \alpha }$ (the subject of trisection), let${\displaystyle A}$ be the point of intersection and let${\displaystyle B}$ be a second point at${\displaystyle {\ell }_2}$. Draw a circle through${\displaystyle B}$ centered at${\displaystyle A}$. (The directrix will be${\displaystyle {\ell }_1}$ and the catch line the circle.) Place the ruler at line${\displaystyle {\ell }_2}$ and mark it at${\displaystyle A}$ and${\displaystyle B}$. Keeping the ruler (but not the mark) touching${\displaystyle B}$, slide and rotate the ruler so that the mark${\displaystyle A}$ touches${\displaystyle {\ell }_1}$, until mark${\displaystyle B}$ again touches the circle. Label this point on the circle${\displaystyle C}$ and let${\displaystyle D}$ be the point where the ruler (and its${\displaystyle A}$-mark) touches${\displaystyle {\ell }_1}$. The angle${\displaystyle \beta =ADB}$ equals one-third of${\displaystyle \alpha }$ (as shown in the visual proof below the illustration of the construction).

Neuseis have been important because they sometimes provide a means to solve geometric problems that are not solvable by means ofcompass and straightedge alone. Examples are thetrisection of any angle in three equal parts, and thedoubling of the cube.^[2][3] Mathematicians such asArchimedes of Syracuse (287–212 BC) andPappus of Alexandria (290–350 AD) freely usedneuseis;Isaac Newton (1642–1726) followed their line of thought, and also used neusis constructions.^[4] Nevertheless, gradually the technique dropped out of use.

In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower offields over${\displaystyle \mathbb{Q}}$,${\displaystyle \mathbb{Q}=K_0\subset K_1\subset \cdots \subset K_n=K}$, such that the degree of the extension at each step is no higher than 6. Of allprime-power polygons below the 128-gon, this is enough to show that the regular23-, 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 regularp-gon is constructible, then${\displaystyle {\zeta }_p=e^{\frac{2\pi i}{p}}}$ is constructible, and in these casesp − 1 has a prime factor higher than 5.) The3-,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 the7-,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 allquintics (fifth-order polynomials) have neusis-constructible roots, which is relevant for the11-, 25-, 31-, 41-, 61-, 101-, and 125-gons.^[5] Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible;^[2] 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 formp = 2^r3^s5^t + 1 wheret > 0 (all prime numbers that are greater than 11 and equal to one more than aregular number that is divisible by 10).^[5]

Neusis can notsquare the circle, as all ratios constructible by neusis arealgebraic, and so can not construct transcendental ratios like${\displaystyle \sqrt{\pi }}$.

T. L. Heath, the historian of mathematics, has suggested that the Greek mathematicianOenopides (c. 440 BC) was the first to put compass-and-straightedge constructions aboveneuseis. The principle to avoidneuseis whenever possible may have been spread byHippocrates of Chios (c. 430 BC), 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 himEuclid too shunnedneuseis in his very influential textbook,The Elements.

The next attack on the neusis came when, from the fourth century BC,Plato'sidealism 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:

1. constructions with straight lines and circles only (compass and straightedge);
2. constructions that in addition to this use conic sections (ellipses,parabolas,hyperbolas);
3. constructions that needed yet other means of construction, for exampleneuseis.

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 mathematicianPappus of Alexandria (c. 325 AD) as "a not inconsiderable error".

• Alhazen's problem
• Angle trisection
• Constructible polygon
• Mathematics of paper folding
• Pierpont prime
• Tomahawk (geometry)
• Trisectrix

• 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).

• MathWorld page
• Angle Trisection by Paper Folding

• Euclidean plane geometry
• Greek mathematics

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