Solve cubic equations using neusis

[GenePeer]

The neusis construction (from Greek neuein = 'incline towards', plural: neuseis) consists of fitting 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 number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.

In an essence, using normal straightedge and compass construction, we can add, subtract, multiply, divide and find a square root. This means that we can find the real roots of a quadratic equation with rational coefficients because the quadratic formula consists of only these operations. The easy and fast way to do this would be http://www.concentric.net/~pvb/ALG/rightpaths.html" .

We can also deduce that we can't solve cubic equations because the cubic formula also includes finding a cubic root. However, with neusis construction we can find the real roots of cubic equations because it can find a cube root. Does anyone know how we could solve the general cubic equation using neusis? Preferably an extension to Lill's Method.

PS: This was my topic for my http://en.wikipedia.org/wiki/Extended_essay" , and I've already handed it in. This is just further (personal) research.

JPM.

 

[HallsofIvy]

Well, I googled on "neusis" and got a number of hits. In particular, this one
http://en.wikipedia.org/wiki/Trisection_of_any_angle
gives a "neusis" construction of the trisection of an angle.

 

[GenePeer]

In my paper I showed how one could solve cubic equations of specific forms like
$$a_{3}x^{3}+a_{2}x^{2}+a_0=0$$ or $$a_{3}x^{3}+a_{1}x+a_0=0$$
which I had to figure out on my own. The only help I got on the internet is finding the cube root (simplest cubic equation). Then I went on to show why neusis should be capable of solving cubic equation where straightedge and compass fail; My first post is basically a summary of that.

I guess I picked a topic people barely investigated. What's really annoying is that neusis is equivalent to origami (art of paper folding), and there are plenty of articles describing how to solve the general cubic equations (even Quartic equations) with origami, but barely any for neusis. I'll look further into after I finish my exams; Life's to hectic now with the final exams on November.