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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematicians.


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 of
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
alone. Examples are the trisection of any angle in three equal parts, and the doubling of the cube. Mathematicians such as
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
of Syracuse (287–212 BC) and
Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
(290–350 AD) freely used ''neuseis''; Sir
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
(1642–1726) followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use.


Regular polygons

In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower of fields over \Q, \Q = K_0 \subset K_1 \subset \dots \subset K_n = K, 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 \zeta_p = e^\frac 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 him
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
too shunned ''neuseis'' in his very influential textbook, '' The Elements''. The next attack on the neusis came when, from the fourth century BC,
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
's
idealism Idealism in philosophy, also known as philosophical realism or metaphysical idealism, is the set of metaphysics, metaphysical perspectives asserting that, most fundamentally, reality is equivalent to mind, Spirit (vital essence), spirit, or ...
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 with straight lines and circles only (compass and straightedge); # constructions that in addition to this use conic sections (
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s,
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
s, hyperbolas); # 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 Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
() as "a not inconsiderable error".


See also

* Alhazen's problem *
Angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
* Constructible polygon * Pierpont prime * Tomahawk (geometry) * Trisectrix


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


MathWorld page

Angle Trisection by Paper Folding
{{Ancient Greek mathematics Euclidean plane geometry Greek mathematics