I'd mentioned two of three problems that mathematicians had been unable to solve by ruler-and-compass methods. Here they are, along with the third one.
- Duplication of the cube: find a cube size that gives twice the volume of some cube. Equivalent to finding 21/3.
- Trisecting the angle: find an angle that is 1/3 of some angle. Equivalent to solving 4*x3 - 3*x = cos(a) for x given a, where x = cos(a/3).
- Squaring the circle: find a square whose area is equal to the area of some circle. Equivalent to finding sqrt(pi).
The first two were shown to be insoluble by that complicated Galois-group method that I had posted on just before this. It's more-or-less that cube roots can't be turned into square roots.
The third one is a result of pi being transcendental, and the proof of that is rather complicated:
Lindemann–Weierstrass theorem - Wikipedia
Also from antiquity is what seems to me like a conjecture of unsolvability. Euclid's fifth postulate or parallel postulate:
Quote:
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If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
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For over 2000 years, that cumbersome-looking axiom stood out like a sore thumb in Euclid's
Elements, with numerous mathematicians trying to prove it from Euclid's other axioms, and failing. A common feature of such attempted proofs was including something equivalent to Euclid's fifth.
But in the mid 19th cy., some mathematicians proved that it was independent of Euclid's other axioms, and that one could substitute some contrary ones without causing inconsistency. That was the beginning of non-Euclidean geometry.