So I did a bit of thinking. There’s a prosthaphaeretic rule that lets you calculate square roots using nothing more than trigonometric functions. Is there one that lets you calculate cube roots?
And I don’t know. I don’t see where there is one. I may be overlooking an approach, though. Let me outline what I’ve thought out.
First is square roots. It’s possible to find the square root of a number between 0 and 1 using arc-cosine and cosine functions. This is done by using a trigonometric identity called the double-angle formula. This formula, normally, you use if you know the cosine of a particular angle named θ and want the cosine of double that angle:
If we suppose the number whose square we want is 
Why can’t I do the same thing with a triple-angle formula? … Well, here’s my choices among the normal trig functions:
Yes, I see you in the corner, hopping up and down and asking about the cosecant. It’s not any better. Trust me.
So you see the problem here. The number whose cube root I want has to be the 
Can it be worked out? Oh, sure, yes. There’s absolutely approximation schemes that would let me find a value of θ which makes true, say,
But: is there a way takes less work than some ordinary method of calculating a cube root? Even if you allow some work to be done by someone else ahead of time, such as by computing a table of trig functions? … If there is, I don’t see it. So there’s another point in favor of logarithms. Finding a cube root using a logarithm table is no harder than finding a square root, or any other root.
If you’re using trig tables, you can find a square root, or a fourth root, or an eighth root. Cube roots, if I’m not missing something, are beyond us. So are, I imagine, fifth roots and sixth roots and seventh roots and so on. I could protest that I have never in my life cared what the seventh root of a thing is, but it would sound like a declaration of sour grapes. Too bad.
If I have missed something, it’s probably obvious. Please go ahead and tell me what it is.
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