An elementary proof of the classic Routh method for counting the number of left half-plane and right half-plane zeroes of a real coefficient polynomial P-n(s) of degree n is given. Such a proof refers to the polynomials P-i(s) of degree i less than or equal to n formed from the entries of the rows of order i and i - 1 of the relevant Routh array. In particular, it is based on the consideration of an auxiliary polynomial (P) over cap(i)(s; q), linearly dependent on a real parameter q, which reduces to either polynomial P-i(s) or to polynomial Pi-1(s) for particular values of q. In this way, it is easy to show that i - 1 zeroes of P-i(s) lie in the same half-plane as the zeroes of Pi-1(s), and the remaining zero lies in the left or in the right half-plane according to the sign of the ratio of the leading coefficients of P-i(s) and Pi-1(s). By successively applying this property to all pairs of polynomials in the sequence, starting from P-0(s) and P-1(s), the standard rule for determining the zero distribution of P-n(s) is immediately derived.