A method is proposed for determining the surface tensions of a solid in contact with either a liquid or a vapor. Only an equilibrium adsorption isotherm at the solid-vapor interface needs to be added to Gibbsian thermodynamics to obtain the expressions for the solid-vapor and the solid-liquid surface tensions, gamma(SV)([1]) and gamma(SL)([1]), respectively. An equilibrium adsorption isotherm relation is formulated that has the essential property of not predicting an infinite amount adsorbed when the pressure is equal to the saturation-vapor pressure. Five different solid-vapor systems from the literature are examined, and found to be well described by the new isotherm relation. The surface-tension expressions obtained from the isotherm relation are examined by determining the surface tension of the solid in the absence of adsorption, gamma(S0)([1]), a material property of a solid surface. The value of gamma(S0)([1]) can be determined by adsorbing different vapors on the same solid, determining the isotherm parameters in each case, and then from the expression for gamma(SV)([1]) taking the limit of the pressure vanishing to determine gamma(S0)([1]). From previously reported measurements of benzene and of n-hexane adsorbing on graphitized carbon, the same value of gamma(S0)([1]) is obtained.