The age-of-the-Earth paradox that beset geology toward the end of the 19th century has been widely misunderstood. Kelvin showed that if Earth were cooling by thermal diffusion, it would have reached its present state after about 20 Myr, far less time than required by sedimentary geology. The assumptions in this calculation were insecure, and the age estimate would have carried little weight if it had not been comparable to that of the Sun, deduced from its gravitational energy. It was the age limit imposed by the available solar power that inhibited serious rethinking of the terrestrial problem. Contrary to popular supposition, even the discovery of radioactivity did not solve the problem because it contributes negligibly to solar energy. The only possible escape from Kelvin's argument would have been the implausible assumption that the Sun was compositionally heterogeneous, with a strong central concentration of elements such as Si and Fe, in spite of being gaseous and presumably convecting. Since the real Earth age paradox was the inability of available solar power to explain the sedimentary record, it was resolved only by the recognition of thermonuclear fusion in the 1930s. Moreover, although Kelvin's terrestrial heat flux calculation appeared relevant when it indicated an age comparable to that of the Sun, it was not necessary to the paradox and so its invalidation did not remove the paradox. ¿ 2000 American Geophysical Union