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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?

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Are electrons considered mechanical? :) – Bruno Joyal Mar 29 '12 at 4:17
You can calculate 5th roots on a slide rule - does that qualify? – Gerry Myerson Mar 29 '12 at 5:29
I'd be interested to see a mechanical device, which every time you push its pedal, it spits out one more decimal digit of $\sqrt{2}.$ – user2468 Mar 29 '12 at 5:34
up vote 5 down vote accepted

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 $\log\,b$.

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Thank you, very interesting references. – Will Mar 29 '12 at 6:50

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