Let \(p(n)\) be the number of partitions of the positive integer \(n\) and let \(p_k(n)\) be the number of partitions of \(n\) into exactly \(k\) summands. The author gives an elementary proof that \(\lim_{n \to \infty} n p(n) \exp\{-\pi(2n/3)^{1/2}\}\) exists and is positive, but does not determine its value (known to be \(48^{-1/2}\)). An elementary determination of the value of the limit was later given by \textit{D. J. Newman} [Am. J. Math. 73, 599--601 (1951); Zbl 0043.04501)].