Did anyone ever build a mechanical device to take fifth roots, or solve general quintics?

Question

This question is from a post from John Baez's blog on, among other things, geometrical constructions. I was hoping someone here might know the answer.

In his post, Baez writes that

Nowadays we realize that if you only have a straightedge, you can only solve linear equations. Adding a compass to your toolkit lets you also take square roots, so you can solve quadratic equations. Adding neusis on top of that lets you take cube roots, which—together with the rest—lets you solve cubic equations. A fourth root is a square root of a square root, so you get those for free, and in fact you can even solve all quartic equations. But you can’t take fifth roots.

Which leads to the question:

Puzzle 5. Did anyone ever build a mechanical gadget that lets you take fifth roots, or maybe even solve general quintics?

Answer 1

As noted in this survey paper (and the references therein), there was no shortage of attempts to build (electro-)mechanical devices for the solution of polynomial equations (not just quintics!); generally, the kinematic/mechanical methods (e.g. the one proposed by A.B. Kempe) were only able to obtain real roots, while the electromechanical methods were also able to obtain complex roots in addition to real ones. As expected, the accuracy is rather shabby compared to what we can obtain with today's methods, but was serviceable enough for the needs of their users.

See also this paper and this short note on a machine by Leonardo Torres based on the so-called "endless spindle" mechanism for computing the quantity $\log(a+b)$ from $\log\,a$ and $\log\,b$.

Answer 2

We do not actually know whether a fifth root can be constructed from its radicand by neusis, nor do we know about quintisection of a given angle.

Both problems involve solutions of irreducible quintic equations, which would be impossible if neusis were merely equivalent to construction aided by conic sections or single-fold origami. But in 2002 Baragar showed that neusis is more powerful and demonstrated this extra power with a geometrical solution to the irreducible quintic equation

$$x^5 − 4x^4 + 2x^3 + 4x^2 + 2x − 6 = 0.$$

Benjamin and Snyder developed a method for solving sextic equations satisfying certain relationships using neusis and showed that one such soluble equation has the form

$$(x-u-2)(x^5+ux^4-4u^2x^3-3u^3x^2+3u^4x+u^5)=0$$

where the quintic factor gives $\dfrac xu=2\cos\left(\dfrac{k\pi}{11}\right)$. The scale factor $u$ solves a cubic equation $u^3+2u^2+2u+2=0$ and thus is marked ruler constructible. Thereby the regular hendecagon is neusis constructible.*

Neither of these constructions sheds any light on fifth root extraction or angle quintisection. Baragar's quintic is not radical-soluble. The equation for the hendecagonal cosines is soluble via complex fifth-root extraction, therefore angle quintisection, but Benjamin and Snyder's method obtains the polygon without using angle quintisection.

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*We know now that the cubic equation for $u$ found by Benjamin and Snyder is not entirely accidental. The tribonacci ratio equation $t^3-t^2-t-1=0$, with roots closely related to Benjamin and Snyder's $u$ factor, is coupled to the original quintic equation through the quadratic Gauss sum of $11$th roots of unity. See https://mathoverflow.net/a/432460/86625 for details.