Prove that the cube cannot be tripled in the sense that star

Prove that the cube cannot be tripled, in the sense that, starting with an edge of a cube of volume 1, an edge of a cube of volume 3 cannot be constructed with straightedge and compass. (b) More generally, prove that the side of a cube with volume a natural number n is constructible if and only if n 3 is a natural number.

Solution

Pappus (c. 350) argued, using completely muddled reasoning, that the three constructions cannot be done by straightedge and compass. But the proof that this is the case requires algebraic ideas that the Greeks did not possess. In particular, the link between geometric figures and algebraic equations, developed in the seventeenth century by Fermat and Descartes, is an essential element of the proof. In 1837,

Wantzel published a proof that the cube cannot be duplicated and the general angle cannot be trisected with straightedge and compass