Thermodynamics is a classical phenomenological theory; it is based on a demonstrably reproducible set of measurements. The greatest advance in this science is associated with James[aut]Joule, James (1818-1889) Prescott Joule. He showed experimentally that different forms of energy can be converted into one another. He also joined forces with many of the other natural philosophers in the mid-19th century, such as William Thomson[aut]Thomson, William (1824-1907) (Lord Kelvin), William Rankine[aut]Rankine, William (1820-1872) and Victor Regnault[aut]Regnault, Victor (1810-1878). This is a great story and the details are discussed in both physical and mathematical terms. Chapter 9 presents the precise experimental work of Regnault. After a sufficient set of experiments established the “Laws of Thermodynamics” in verbal form, the time had come for a complete mathematical theory. Chapters 10 , 11 and 12 present and discuss the evolution of the mathematical theory associated with Rankine, Thomson (Kelvin) and Clausius.