This paper presents new measurements of the viscosity of gaseous R407C (23 mass% HFC-32, 25 mass% HFC-125, 52 mass% HFC-143a) and R407E (25 mass% HFC-32, 15 mass% HFC-125, 60 mass% HFC-143a). The measurements were carried out with an oscillating-disk viscometer of the Maxwell type at temperatures from 298.15 to 423.15 K. The densities of these two fluid mixtures were calculated with the equation-of-state model in REFPROP. The viscosity at normal pressures was analyzed with the extended law of corresponding states developed by Kestin et al., and the scaling parameters needed in the analysis were obtained from our previous studies for the viscosity of the binary mixtures consisting of HFC-32, HFC-125, and HFC-134a. The modified Enskog theory developed by Vesovic and Wakeham (V-W method) was applied to predict the viscosity for the ternary gaseous HFC mixtures under pressure. As for the calculation of pseudo-radial distribution functions in mixtures, a method based on the equation of state for hard-sphere fluid mixtures proposed by Carnahan-Starling was applied. It was found that the V-W method can predict the viscosity of R407C and R407E without any additional parameters for the ternary mixture.