124 BELL SYSTEM TECHNICAL JOURNAL 



It is evident that the equation is a comparison between two states. Do 

 not go astray by supposing that these are like two of the states which we 

 have been considering, having the same V and V and differing in the number 

 of inventories! These on the contrary are two examples of THE STATE — 

 of the thermodynamic state, of the most probable state — of a gas, differing 

 in the values of some at least among the five variables. The quantities dU 

 and dS and dV are the differences between the Z^-values and the ^-values and 

 the F-values of the two states, while P and T may be taken as referring to 

 either, the smallness of the difference between the two states — implied by 

 the differential notation — permitting of this. 



It is also evident from my wording that the one equation is being used 

 to define the two quantities 5 and T. This is unluckily no verbal slip, 

 nor is it a temporary shortcut to be replaced by a royal road as the ar- 

 gument proceeds. The meanings of entropy and temperature are so 

 coiled up together in thermodynamics, that it is impossible to take them 

 apart unmutilated. One cannot seize either by storm and then invest the 

 other, at least not without the aid of statistical theory' : one has to surround 

 them both in a single campaign. As Eddington has vigorously written, this 

 is a common thing in physics. Electric force is defined as that which acts on 

 electric charge, electric charge as that which is acted upon by electric force, 

 and so on. . . . Common as it may be, it is probably nowhere else so har- 

 assing as in thermodynamics. There are three ways of intruding upon the 

 vicious circle. 



First, to apprehend both concepts in a single mental act. This is the 

 counsel of perfection. 



Second, to use a temporary definition of temperature, with the promise of 

 confirming or correcting it later. The ideal-gas thermometer is the device 

 used for this purpose in thermodynamics. Anyone trained in this way is 

 likely to think for the rest of his life of temperature as the primary concept, 

 entropy as a derived one — as indeed was the case, when thermodynamics 

 started. 



Third, to produce a theory which makes a pronouncement as to the nature 

 of entropy. 



This last is the major office of statistical mechanics. To those who accept 

 it, entropy becomes the primary concept and temperature the derived one, 

 and both are visualized by the aid of the key-word "probability" of the basic 

 theorem, interpreted in some particular way. 



Old Statistical Theory of Entropy 



In the classical statistics, the entropy of a distribution is considered to be the 

 logarithm of the number of inventories or complexions compatible with thatdis- 



