Percy writes:
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. True, one way of trisecting an angle requires taking a cube root, but is that the only way, and can it be proved?
During plane geometry in the 8th grade, construction component, we learned how to bisect any angle and demonstrate the proof; the teacher told us it was impossible to trisect an angle with construction, and merely replied to my question that the math that proved this was beyond my comprehension.
So I set about trisecting an angle with construction. Many weeks and reams of paper later, I came back with three methods: one for a 90 degree angle, one for acute angles, and one for obtuse angles. He pointed out that my solution required knowing which category any given angle belonged to and so was no solution at all. I still refused to accept the impossibility, though I stopped my mad nights with compass and protractor. I did learn a great deal about plane geometry in the process, though, and appreciated the difficulty as a matter of converting "odds" to "evens"...
The difficulty of our increasingly specialized world goes beyond mathematics: we are encouraged to take part in medical decisions, but our knowledge can never equal that of the physicians asking us to make those decisions; we necessarily put our lives in the hands of the engineers who design our vehicles, ground and air, the workers who build them, and the mechanics who repair and maintain them. The list is as long as the roster of specialized roles in a technological society.
The best we can do, I think, is to maintain a healthy skepticism and to examine past results and the opinions of the experts' peers: the aeronautical engineer who claims her radical half-wing design is perfectly safe has to be judged in light of the opinions of other engineers and the track record of her invention. The real danger lies in accepting the first "credentialed" opinion one encounters.
I never completely accepted the impossibility of trisecting an angle, because it was never demonstrated to me, mostly because my six semesters of college math stopped after calculus and analytic geometry, though in fact I never asked about it again.
I'm comfortable living with my uninformed skepticism for two reasons: the cost of informing that skepticism is high, and the relevance of the truth of the matter to my daily life is low. I suppose we all negotiate those parameters.
Edited by Omnivorous, : topy
Drinking when we are not thirsty and making love at any time, madam, is all that distinguishes us from the other animals.
-Pierre De Beaumarchais (1732-1799)
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