Neusis construction

[1][2] Mathematicians such as Archimedes of Syracuse (287–212 BC) and Pappus of Alexandria (290–350 AD) freely used neuseis; Sir Isaac Newton (1642–1726) followed their line of thought, and also used neusis constructions.

In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower of fields over

[4] Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible;[1] the 25-, 31-, 41-, 61-, 101-, and 125-gons remain open problems.

[4] T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (c. 440 BC) 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 (c. 430 BC), 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.

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 (c. 325 AD) as "a not inconsiderable error".