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 from equations contained in his first paper. His proof of the second law is not free from objections. In March, 1851, appeared a paper of William Thomson which contained a perfectly rigorous proof of the second law. He obtained it before he had seen the researches of Clausius. The statement of this law, as given by Clausius, has been much criticised, particularly by Rankine, Theodor Wand, P. G. Tait, and Tolver Preston. Repeated efforts to deduce it from general mechanical principles have remained fruitless. The science of thermodynamics was developed with great success by Thomson, Clausius, and Rankine. As early as 1852 Thomson discovered the law of the dissipation of energy, deduced at a later period also by Clausius. The latter designated the non-transformable energy by the name entropy, and then stated that the entropy of the universe tends toward a maximum. For entropy Rankine used the term thermodynamic function. Thermodynamic investigations have been carried on also by G. Ad. Hirn of Colmar, and Helmholtz (monocyclic and polycyclic systems). Valuable graphic methods for the study of thermodynamic relations were devised in 1873–1878 by J. Willard Gibbs of Yale College. Gibbs first gives an account of the advantages of using various pairs of the five fundamental thermodynamic quantities for graphical representation, then discusses the entropy-temperature and entropy-volume diagrams, and the volume-energy-entropy surface (described in Maxwell's Theory of Heat). Gibbs formulated the energy-entropy criterion of equilibrium and stability, and expressed it in a form applicable to complicated problems of dissociation. Important works on thermodynamics have been prepared by Clausius in 1875, by R. Rühlmann in 1875, and by Poincaré in 1892.

In the study of the law of dissipation of energy and the principle of least action, mathematics and metaphysics met on