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Why can a thermometer serve as a third system in the zeroth law of thermodynamics?

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Engraved portrait of Rudolf Julius Emanuel Clausius, a bearded man in formal attire with sideburns, facing forward. — thermodynamics Rudolf Clausius (1822 - 1888) is considered one of the central founders of the science of thermodynamics.

The zeroth law of thermodynamics states that if two systems are each in thermal equilibrium with a third system, they must also be in thermal equilibrium with each other.12

A thermometer serves as this "third system" because it allows for an objective measurement of temperature. When a thermometer is brought into thermal contact with an object, it reaches thermal equilibrium with that object, meaning both are at the same temperature. Because the thermometer’s physical properties—such as its volume or electrical resistance—change in a predictable way with temperature, it can be used to indicate the temperature of the system it is measuring.31

By extension, if two different objects are both in thermal equilibrium with the same thermometer, the zeroth law guarantees that those two objects are in thermal equilibrium with each other, effectively sharing the same temperature. This principle provides the logical foundation for defining temperature scales and using thermometers as reliable diagnostic tools.132