[1] In a paper from 1837,[2] Wantzel proved that the problems of are impossible to solve if one uses only a compass and straightedge.
[5] Wantzel was also the first person to prove, in 1843,[6] that if a cubic polynomial with rational coefficients has three real roots but is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone; that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals.
This theorem would be rediscovered decades later by (and sometimes attributed to) Vincenzo Mollame and Otto Hölder.
Ordinarily he worked evenings, not lying down until late; then he read, and took only a few hours of troubled sleep, making alternately wrong use of coffee and opium, and taking his meals at irregular hours until he was married.
He put unlimited trust in his constitution, very strong by nature, which he taunted at pleasure by all sorts of abuse.