The first statement of the true dynamical theory of heat, based on the fundamental idea that the work done in a Carnot cycle is to be accounted for by an excess of the heat received from the source over the heat delivered to the refrigerator, was given by Clausius in a paper which appeared in _Poggendorff's Annalen_ in March and April 1850, and in the _Philosophical Magazine_ for July 1850, under a title which is a German translation of that of Carnot's essay. In that paper the First Law of Thermodynamics is explicitly stated as follows: "In all cases in which work is produced by the agency of heat, a quantity of heat proportional to the amount of work produced is expended, and, inversely, by the expenditure of that amount of work exactly the same amount of heat is generated." Modern thermodynamics is based on this principle and on the so-called Second Law of Thermodynamics; which is, however, variously stated by different authors. According to Clausius, who used in his paper an argument like that of Carnot based on the transference of heat from the source to the refrigerator, the foundation of the second law was the fact that heat tends to pass from hotter to colder bodies. In 1854 (_Pogg. Ann._, Dec. 1854) he stated his fundamental principle explicitly in the form: "Heat can never pass from a colder to a hotter body, unless some other change, connected therewith, take place at the same time," and gives in a note the shorter statement, which he regards as equivalent: "Heat cannot of itself pass from a colder to a hotter body." We shall not here discuss the manner in which Clausius applied this principle: but he arrived at and described in his paper many important results, of which he must therefore be regarded as the primary discoverer. His theory as originally set forth was lacking in clearness and simplicity, and was much improved by additions made to it on its republication, in 1864, with other memoirs on the Theory of Heat. In the _Transactions R.S.E._, for March 1851, Thomson published his great paper, "On the Dynamical Theory of Heat." The object of the paper was stated to be threefold: (1) To show what modifications of Carnot's conclusions are required, when the dynamical theory is adopted: (2) To indicate the significance in this theory of the numerical results deduced from Regnault's observations on steam: (3) To point out certain remarkable relations connecting the physical properties of all substances established by reasoning analogous to that of Carnot, but founded on the dynamical theory.