The conceptual import of Carnot's theorem to the discovery of the entropy

Summary Among many other things, Carnot stated a principle and proved a theorem. In 1850, Clausius corrected Carnot's theory, modifying it according to Joule's principle. He might have considered a corollary of the theorem as the mathematical formulation of Carnot's principle. We challenge the corollary: it is based on hidden assumptions, nor is it true for all cycles. Clausius realized the corollary's lack of generality, but on different grounds. In 1854, he generalized the theorem, and gave an (other) expression to Carnot's principle. We analyze Clapeyron's account of Carnot's theory, Thomson's account of 1849 and some of Clausius belated comments on his 1850 paper, as well Clausius' paper of 1854. We hope that they shed light on the corollary's tacit hypotheses and on the meaning of Carnot's principle. It is our contention: Clausius took seriously a contemporary meaning of the principle, and looked for a condition of integrability that could express recovery of the initial conditions of the reservoirs. Furthermore, he seems to have had some prior knowledge of the form the expression of the principle should take. Actually, this was the theory's natural candidate.