Worked solution: Wantzel's algebraic impossibility proof for doubling the cube (1837)

Menaechmus (c. 350 BCE) is credited with the earliest known solution to doubling the cube, using conic sections. Take the two parabolas $x^2=y$ and $y^2=2x$; substituting $y=x^2$ from the first into the second gives $x^4=2x$, i.e. $x^3=2$ for $x\neq0$. Their intersection point therefore has $x$-coordinate exactly $\sqrt[3]{2}$ — but parabolas cannot be drawn with an unmarked straightedge and compass, so this does not conflict with Wantzel's later proof; it answers a different question, one where more curves are allowed as tools.

Other ancient solutions include Archytas's remarkable three-dimensional curve (the intersection of a cylinder, a cone, and a torus, reconstructed in modern accounts of his work) and, like the trisection problem, solutions by neusis (marked ruler), which Wantzel's paper does not address. All of these predate Wantzel by two millennia; what changed in 1837 was not finding a new construction but proving, for the first time, that none using only straightedge and compass could exist.

Wantzel handled doubling the cube and angle trisection in the same 1837 paper, using the same core idea — degrees of field extensions must be powers of $2$ — and this became, alongside his completion of Gauss's polygon theorem, one of the earliest applications of field-theoretic reasoning to settle a purely geometric question, a few years before Galois's ideas on field extensions and solvability became widely known.