Neusis construction

The **neusis** is a geometric method that was used by ancient Greek mathematicians. At that time, people did not have good rules to measure distances. They did not know algebra. So, they studied geometry by moving fixed length sticks around on a piece of paper.

## Geometric construction

The neusis construction (from Greek νεῦσις from νεύειν _neuein_ "incline towards"; plural: νεύσεις _neuseis_) fits a line element of given length (_a_) in between two given lines (_l_ and _m_), in such a way that the line element, or its extension, passes through a given point _P_. That is, one end of the line element has to lie on _l_, the other end on _m_, while the line element is "inclined" towards _P_.

A neusis construction might be performed by means of a 'Neusis Ruler': 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 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 _a_ from the origin. The yellow eye is moved along line _l_, until the blue eye coincides with line _m_. The position of the line element thus found is shown in the figure as a dark blue bar.

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_ (διάστημα; Greek for "distance").

## 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, or the construction of a regular heptagon. Mathematicians such as Archimedes of Syracuse (287–212 BC) freely used _neuseis_. Slowly, the Greeks stopped using it.

## Decline in use

T. L. Heath, the historian of mathematics, thought that the Greek mathematician Oenopides (ca. 440 BC) was the first to put compass and straightedge constructions above _neuseis_. Hippocrates of Chios (ca. 430 BC) may have spread the idea that mathematicians should avoid _neuseis_ whenever possible. He came from the same island as Oenopides, and was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him, Euclid also said bad things about _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 groups were:
1. constructions with straight lines and circles only (compass and straightedge);
1. constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas);
1. constructions that needed yet other means of construction, for example _neuseis_.

In the end, the Greek mathematicians thought that using a neusis was acceptable only when the two other, higher groups 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 (ca. 325 AD) as "a not inconsiderable error".

## Related pages

- Angle trisection
- Doubling the cube
- Squaring the circle

## 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).

## Other websites

- MathWorld page
- Angle Trisection by Paper Folding