The part of his theorem to which I am referring is the equation relating the heat & temperature for a reversible between temperature sources. QH / TH = - QC / TC Did he derive somehow from the forerunner equations of the ideal gas law (i.e., Charles & Boyle's Laws)? (He could not have taken it from Clapeyron's ideal gas law since he had already died by the time that law got forumlated!) Or did he formulate it from observing real steam engines? Or did he somehow do it another way? From this equation, it is pretty easy to figure out how to express a property that measures the irreversibility of heat transfer (i.e., entropy) by simply making this property be constant for a reversible process - which this equation does even for a differential amount of heat transfer since such reversible heat transfer is done as a quasi-static process in which the temperature is the same (i.e., the cycle fluid & the temperature source) throughout the process. So the key to figuring out how entropy is derived is understanding how Carnot figured out this equation!