Well, the question is, what is directly measurable, and I don't see how any non-mechanical thermodynamic parameter is directly measurable. You have to have something linear with ticks or tick marks on it and make a direct comparison and read off a scale. A ruler, a protractor, a clock. Even measuring mass is a geometric thing, use a balance and read an angle of deflection. Force you measure mass and acceleration, and for acceleration you use a ruler and a clock. It's almost like the only directly measurable thing is space and time. I think Lieb and Yngvason's point was that IF you had a machine that answered yes or no as to whether one system was adiabatically accessible from another, you would have an "entropy ruler" by which you could directly measure entropy. No need to impose a quasistatic restriction, or introduce heat or temperature or internal energy. Caratheodory's exposition is vague to me, only because I have not studied it enough, I am sure. As I understand it, his exposition is also based on adiabatic accessibility. Lieb and Yngvason have given me a better grasp of adiabatic accessibility, maybe now I can start to make better sense of Caratheodory. Thanks for that Caratheodory reference, maybe that is where I should begin.