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Temperature measurements started with the developments of thermoscopes and thermometers. They were based on the fact that liquids in general expanded significantly on change in temperature (especially mercury). The initial mercury thermometers were calibrated against common physical phenomenon and divided linearly (for example, the Celcius scale).

But now we know that materials do not expand linearly with temperature. How was this knowledge incorporated to get a material independent temperature scale, because if we continued using a linear scale we would face problems at high temperatures?

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  • $\begingroup$ We use other phenomenas than thermal dillatation of liquids, for example effects in semiconductors where we know preciselly how some parameter is dependent on temperature. $\endgroup$ – Martin Vesely Apr 28 at 22:32
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Fundamentally, as the field of Statistical Mechanics (aka Thermodynamics) developed, it became possible to define temperature based on fundamental units and constants. As you probably know, temperature in general is defined as something proportional to the logarithm of the number of available energy states in the system. To keep people's brains from exploding, the numeric values were 'adjusted' so that, e.g., zero Celsius could stay where it was originally defined (freezing point at standard pressure).

So, short answer is that the rigorous mathematical definitions of temperature are based on quantum energy levels as well as phonons, etc., and the final numerical values were chosen to align with the then-current (but slightly inaccurate) values for various phase transitions.

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