A new method is presented for the isolation of the real roots of a given integral, univariate, square-free polynomial P. This method is based on Vincent's theorem and only uses: (i) Descartes' rule of signs, and (ii) transformations of the form x = a1 + 1/x′, x′ = a2 + 1/x″, x″ = a3 + 1/x‴, ..., for positive, integral ai's. The key element in this procedure is the calculation of the quantities a1, a2, a3,... . We compute them as "positive lower root bounds" of polynomials and the resulting algorithm has the best theoretical computing time achieved thus far. Empirical results also verify the superiority of our method over all others existing.