neusis

Neusis is a method in classical geometry for constructing a desired point or length using a straightedge marked with a fixed segment. The term comes from the Greek neusis, meaning “finding” or “taking up.” In a neusis construction, a line is drawn and a ruler marked with a fixed length is allowed to slide along that line. The ruler is positioned so that one end of the marked segment moves on the line while a chosen point on the ruler passes through a predetermined point or lies on a given curve. The intersection thus determined yields the construction. This approach extends the toolkit beyond ordinary straightedge and compass by using the fixed mark as an additional constraint.

In a typical neusis setup, the marked ruler slides along a base line while a specific point on the ruler is constrained to pass through a target point or meet a given curve (such as a circle). The accepted length of the marked segment is fixed in advance, and the geometry of the arrangement determines the desired construct.

Neusis constructions can solve certain problems that are not achievable with unmarked straightedge and compass alone. They are capable of performing tasks such as trisecting arbitrary angles and solving certain cubic equations, including constructions related to cube roots, within the marked-ruler framework. Because neusis requires a marked ruler, it is not part of the classical Euclidean toolkit, but it is studied as an extension of constructibility in the broader context of geometry.