3. Physical Proof that the Geometric Mean of any Number of Quantities is less than the Arithmetic Mean

Abstract

If a number of equal masses of the same material be given, at different temperatures, and enclosed in an envelope impervious to heat, they will finally assume a common temperature; which is the arithmetic mean of the initial temperatures, if the material be one whose specific heat does not vary with temperature.But they may be brought to a common temperature by means of reversible thermodynamic engines employed to obtain the utmost amount of work from the initial unequal distribution. This question was first investigated by Thomson (Phil. Mag. 1853, “On the Restoration of Energy from an unequally heated Space”), and the application of his method to the present problem shows that the final common temperature of the masses, when as much work as possible has been obtained from them, is the geometric mean of the initial temperatures; but this investigation introduces the condition that the temperatures must be measured from the absolute zero.