Two nucleation theorems are proved using small system thermodynamics. The first is well known and has been used before to determine the number of molecules in the critical nucleus controlling the nucleation of droplets from supersaturated vapours. The second appears to be new, and relates the temperature dependence of the nucleation rate to the excess internal energy of the critical cluster. An analysis of measured homogeneous nucleation rates can therefore provide the size, internal energy, free energy, and entropy of the critical cluster, which should provide important guidance for the construction of theoretical models of the process. This is illustrated using water, Ir-butanol and n-nonane nucleation data. While there is often a close correspondence between the droplet free energy and the form suggested by classical theory, the excess internal energy seems to show a linear dependence on molecular number, rather than the classical 2/3 power behaviour.