Hayes goes on to explain why trisecting angles is impossible. I'd never seen it explained before, and he tells us that it is explained very infrequently. It turns out it's impossible because you can do square roots and 4th roots and 8th roots using geometry, but not cube roots. To trisect an angle you need to use geometry to compute a cube root, something known to be impossible.
But, I would ask him, how does he know that taking a cube root is the only approach to trisecting an angle.
That's not quite the argument. The argument is that if you had any straightedge-and-compasses method of trisecting the angle, then you could use this method to find non-rational roots of certain cubic equations (the author of your article instances 8u
3 - 6u = 1). But this is impossible to do using straightedge and compasses, so you can't trisect the angle. It's not: "To trisect an angle you need to use geometry to compute the root of a cubic equation" but: "If you could trisect an angle then you
would be able to use geometry to compute the root of a cubic equation".
Edited by Dr Adequate, : No reason given.
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