In geometry, the neusis (νεῦσις; from Ancient Greek νεύειν (neuein) 'incline towards'; plural: νεύσεις, neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians.
Geometric construction The neusis construction consists of fitting a straight line element of given length (a) in between two given (not necessarily straight) lines (l and m), in such a way that the extension of the line element passes through a given point P. That is, one end of the line element has to lie on l and the other end on m while the line element is "inclined" towards P. 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 (Greek: διάστημα, lit. 'distance'). A neusis construction might be performed by means of 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; 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. If we require both lines l and m 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.
Trisection of an angle by line–circle neusis
Starting with two lines ℓ 1 {\displaystyle \ell _{1}} and ℓ 2 {\displaystyle \ell _{2}} that intersect at angle α {\displaystyle \alpha } (the subject of trisection), let A {\displaystyle A} be the point of intersection and let B {\displaystyle B} be a second point at ℓ 2 {\displaystyle \ell _{2}} . Draw a circle through B {\displaystyle B} centered at A {\displaystyle A} . (The directrix will be ℓ 1 {\displaystyle \ell _{1}} and the catch line the circle.) Place the ruler at line ℓ 2 {\displaystyle \ell _{2}} and mark it at A {\displaystyle A} and B {\displaystyle B} . Keeping the ruler (but not the mark) touching B {\displaystyle B} , slide and rotate the ruler so that the mark A {\displaystyle A} touches ℓ 1 {\displaystyle \ell _{1}} , until mark B {\displaystyle B} again touches the circle. Label this point on the circle C {\displaystyle C} and let D {\displaystyle D} be the point where the ruler (and its A {\displaystyle A} -mark) touches ℓ 1 {\displaystyle \ell _{1}} . The angle β = A D B {\displaystyle \beta =ADB} equals one-third of α {\displaystyle \alpha } (as shown in the visual proof below the illustration of the construction).
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, and the doubling of the cube. Mathematicians such as Archimedes of Syracuse (287–212 BC) and Pappus of Alexandria (290–350 AD) freely used neuseis; Isaac Newton (1642–1726) followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use.
Regular polygons Suppose that a number x to be constructed lies in a tower of fields over Q {\displaystyle \mathbb {Q} } ,
Q = K 0 ⊂ K 1 ⊂ ⋯ ⊂ K n = K . {\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K.}
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