text: The original definition of Carathéodory was limited to reversible, quasistatic process, described by a curve in the manifold of equilibrium states of the system under consideration. He called such a state change adiabatic if the infinitesimal heat differential form  
  vanishes along the curve.  In other words, at no time in the process does heat enter or leave the system. Carathéodory's formulation of the Second Law of Thermodynamics then takes the form: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." From this principle he derived the existence of entropy as a state function 
  whose differential  is proportional to the heat differential form  , so it remains constant under adiabatic state changes (in Carathéodory's sense). The increase of entropy during irreversible 
  processes is not obvious in this formulation, without further assumptions.