Neusis construction

Example trisection of the angle
The radius of the semicircle is equal to the marked segment of the ruler, the length of the drawn line corresponds to the segment ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\overline {CD}}}$ ${\displaystyle {\overline {BD}}.}$

The Neusis construction (from the Greek Neusis for inclination), in the English-speaking area Neusis construction or verging construction , is a geometric construction method using the so-called insertion ( Neusis ) . This means drawing in a straight line using a ruler, on which the length of a desired route is determined by two permanently attached markings.

The ruler is placed on the drawing sheet in the relevant construction step and brought into the functional position. Then draw a line with the specified marked length along its edge.

The Neusis construction enables geometrical tasks to be solved exactly which, as a construction with compasses and ruler, do not provide a solution, such as B. Trisection of the angle , doubling of the cube , squaring the circle and heptagon . According to Bartel Leendert van der Waerden , the Neusis shows the falsity of the view that ancient Greek mathematics only allowed constructions with compasses and rulers; Pappos even explicitly refers to the use of the Neusis for tasks that cannot be solved with compasses and ruler.

Neusis constructions are already known from antiquity. Famous users were u. a. Hippocrates of Chios (5th century BC), who used it to determine the area of ​​his little moons, Archimedes of Syracuse (3rd century BC), who used it to construct the regular heptagon (heptagon after Archimedes ) and with a Neusis ruler and a circle, Nicomedes , who used it to construct his conchoid of Nicomedes , with which he doubled the cube and divided the angle into three , Pappos of Alexandria (in the 4th century AD), who in his mathematical Collection showed that a Neusis construction by Archimedes can be reduced to the intersection of two circles, Apollonios von Perge , in a work on Neusis that has only survived in fragments, in which he shows that some Neusis constructions can be carried out with compasses and ruler, and Abu l-Wafa (990 AD), in his book on geometric constructions.

literature

• Menso Folkerts : Neusis , in: Der Neue Pauly , Brill 2006
• Bartel Leendert van der Waerden : Awakening Science , Volume 1, Birkhäuser, 2nd edition 1966, English edition Kluwer, 5th edition 1988

See also

• Mathematics in the heyday of Islam, Euclidean geometry

Individual evidence

1. ^ Weisstein, Eric W. "Neusis Construction." From MathWorld, A Wolfram Web Resource. accessed on September 15, 2018.
2. ↑ Bodo v. Pape: Macro-Mathematics School Mathematics on New Paths Beyond Algebra and Analysis: Algorithms; BoD-Books on Demand, Nordertedt 2016, p. 388. Page 127 ff. 7.1 Neusis ( excerpt (Google) ), accessed on September 15, 2018.
3. ↑ Van der Waerden, Science Awakening, Kluwer 1988, p. 263
4. ^ Van der Waerden, Science Awakening, p. 132
5. ↑ van der Waerden, Science Awakening, p. 226. Only received in an Arabic manuscript.
6. ↑ Represented in John Horton Conway , Richard K. Guy , The Book of Numbers, Springer 1996, p. 195. There the Neusis construction of the heptagon is sketched with two straight lines.
7. ↑ Van der Waerden, Science Awakening, p. 235
8. ↑ van der Waerden, Science Awakening, p. 286. He also showed that the Neusis construction for the three-dimensional division of Nicomedes can be reduced to the intersection of a circle with a hyperbola. Van der Waerden, Science Awakening, p. 236
9. ^ Van der Waerden, Science Awakening, 263