The continued fraction methods for solving a polynomial equation consist essentially of transforming a polynomial by using x = a + 1/x′, (for instance, to isolate the roots by reducing the number of sign variations). Thus, in stages, a continued fraction expansion for a root emerges. The author strongly advocates the recognition of the priority of A. J. H. Vincent [1] over J. V. Uspensky [2], and he shows the computational and theoretical superiority of Vincent’s work, incidentally offering improvements of his own [3].