In presenting the geometric approach based on the work of Boyling this chapter provides an alternative to the algebraic approach in Chapters 2 to 9. It is well-known that the use of Carathéodory’s version of the Second Law to derive the structure of thermodynamics suffers from the limitation imposed by the local nature of his theorem. As we show, this is overcome in Boyling’s derivation of a global empirical entropy. Of particular relevance is that this account is based on axioms which can be systematically compared with those of Lieb and Yngvason. Once this empirical entropy has been established, it, together with the Zeroth Law, can be used to establish the existence of an empirical temperature with empirical entropy and temperature yielding thermodynamic entropy and temperature. It is shown that the consequential thermodynamic differential 1-form gives rise to a contact structure. In the case of both entropy and internal energy it is found that there are the two possibilities of monotonic increase or decrease in an adiabatic process. It is shown that there is a natural extension of the analysis to systems in which mass is taken as a variable, rather than a parameter, so that scaling and extensivity are satisfied.