We give a geometric method to count the fixed points of an automorphism of an algebraic curve. For any projective morphism of a non-singular irreducible complete algebraic curve, we obtain an inequality involving the number of fixed points of the automorphism, the Weierstrass order- sequence associated to the morphism and the orders of the automorphism with respect to the morphism. This inequality will give us an upper bound for the number of fixed points of the automorphism. This upper bound improves the classical ones, due to Hurwitz, in several cases.