Page "Euclidean geometry" Paragraph 83

Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry.

For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools.

However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible.

Other constructions that were proved impossible include doubling the cube and squaring the circle.

In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve first-and second-order equations, while doubling a cube requires the solution of a third-order equation.