A brief survey of the development of Equilibrium theory, from Carnot’s theorem to the least action principle and the Ergodic Hypothesis (EH), as a background for the study of stationary Non Equilibrium. Particular attention is devoted to the application of the EH and of the Least Action principle to Boltzmann’s derivation of the second law 1866, (presented in Clausius’ version in the new Sect. 1.4). Identification of a thermodynamic equilibrium state with the collection of time averages of observables was made, almost without explicit comments and the analysis relied on EH as the assumption that motions are periodic. A first attempt, already in 1868, to eliminate such a hypothesis appears initially in the form: the molecule goes through all possible states because of the collisions with the others. However the previous hypothesis (i.e. periodic motion covers the energy surface) appears again to use EH, as a starting point, to obtain the microcanonical distribution for the entire gas. The “solution of a mechanical problem” arises in this context to present an example of ergodicity, and it is interesting to examine its conflict (conjectured in the in the first edition, and here developed in Sect.  6.3 ) with KAM theory. Little later (1971) the ergodic hypothesis relation to periodic orbits plays a minor role, and a general theory of the “ensembles” is discovered simultaneously with the equivalence between canonical and microcanonical distributions for large systems, as explicitly recognized by Gibbs. And soon the discovery of H-theorem and Boltzmann’s equation systematized the early work of Maxwell on the theory of gases. However in this book the theme of the Boltzmann equation is not really touched: attention is concentrated on stationary states in and out of equilibrium, with the single exception of the attempt to introduce the “time scale” of a process of transformation of an equilibrium state into another aimed at quantifying the notion of “infinitely slow” (thermodynamically reversible) process in Sect.  5.2 .