The role that the determinants of real, symmetric, Toeplitz matrices play in testing for their positive semidefiniteness is discussed. It is shown that the leading principal minor test is not sufficient in general to test for the positive semidifiniteness of Toepliz matrices, except in certain cases. Several properties of Toeplitz determinants are derived, and the conditions under which the leading principal minor test is indeed sufficient are shown. Because of the special structure of Toeplitz matrices, the author is able to derive a very simple and general test for positive semidefiniteness which does not require the computation of all principal mirrors. >