In geometry, the neusis (Greek, Ancient (to 1453);: νεῦσις; ; plural: grc|νεύσεις|neuseis|label=none) is a geometric construction method that was used in antiquity by Greek mathematicians.
The neusis construction consists of fitting a straight line element of given length in between two given (not necessarily straight) lines (and), in such a way that the extension of the line element passes through a given point . That is, one end of the line element has to lie on and 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 (el|διάστημα|lit=distance).
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; 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 .
If we require both lines and 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
\ell1
\ell2
\alpha
A
B
\ell2
B
A
\ell1
\ell2
A
B
B
A
\ell1
B
C
D
A
\ell1
\beta=ADB
\alpha
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.[2] 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.[3] 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 of fields over
\Q
\Q=K0\subsetK1\subset...\subsetKn=K
\zetap=
| ||||
| e |
Neusis can not square the circle, as all ratios constructible by neusis are algebraic, and so can not construct transcendental ratios like
\sqrt{\pi}
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 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:
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".
