A simple neusis (marked ruler) construction of $\sqrt[3]{2}$ is given in many places, for example wikipedia.
My question is: what is the history of this construction? As far as I can determine, all the ancient Greek constructions used either unmarked straight-edge, or a curve such as Nicomedes' conchoid or Diocles' cissoid. Wikipedia references Dörrie's 100 Great Problems of Elementary Mathematics, but Dörrie himself simply says that
The simplest and most accurate method of obtaining $\sqrt[3]{2}$ is by paper-strip construction.
In other words, using a neusis. I don't know, however, if this construction originated with Dörrie or precedes him.
Does anybody know?
https://www.amazon.com/dp/0486240738
Nicomedes managed to double the cube with neusis, and also described the conchoid of Nicomedes.
https://mathworld.wolfram.com/ConchoidofNicomedes.html
" It is the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive)."
In 2002, Arthur Bartagar proved that neusis and conchoid-assisted constructions were mathematically equivalent.
https://baragar.faculty.unlv.edu/papers/TwiceNotch.pdf
He also proved that at least some irreducible rational quintics can be be solved with neusis/conchoids. (Note that the general equation to find the intersections of a circle with a conchoid of Nicomedes is a sextic polynomial equation)
It is known that neusis/conchoids can not solve irreducible rational septics, nor polynomials irreducible over the rationals whose degree has a prime factor of 7 or more. Whether all sextics and quintics can be solved by this method remains an open question.