A thermometer is a temperature proxy, usually with a fixed number of parameters. When a nearly ideal gas or a similarly simple system is not available, the determination of a temperature scale is difficult, and has been subject to debate. Entropy uniqueness, due to Clausius, provides a fundamental theoretical criterion to answer this question. By measuring the heat flow dQ around a grid in p–V space, with M elemental areas, one can test a proposed temperature scale Tn by computing , the rms deviation, from the desired value of zero. This approach also can be used to determine which of a group of n thermometers has the most accurate temperature scale Tn. As background we discuss Clausius’s development of the mathematics and the concept of entropy. An shows how energy conservation permits one to eliminate the zigs and zags that appear in the Clausius construction for a general Carnot cycle.