Summary.: We show how Liouville’s formulas for the number of representations of a positive integer by the forms (Formula presented.), and (Formula presented.) follow in a simple systematic way from a beautiful identity of Jacobi using some elementary relationships between the infinite series P(x) = 1 + 2x + 2x 4 + 2x 9 + ... and Q(x) = 1 − 2x + 2x 4 + 2x 9 + ... given by Gauss.