relation does not guarantee an equivalence relationship. For the above statement to be true, both reflexivity and symmetry must be implicitly assumed.

It is the Euclidean relationships which apply directly to thermometry. An ideal thermometer is a thermometer which does not measurably change the state of the system it is measuring. Assuming that the unchanging reading of an ideal thermometer is a valid tagging system for the equivalence classes of a set of equilibrated thermodynamic systems,