How do we know that temperature is the random motion of particles of matter? It is written in many textbooks that heat is the random motion of particles but no text book mentions that how do we know that heat is the random motion of particles 
 A: Historically the first field of theoretical physics based on the concept that heat is motion of molecules is kinetic theory of gases
The predecessor of the kinetic theory was caloric theory
According to the wikipedia article in 1857 Rudolf Clausius developed a theory that included rotational and vibrational modes of molecules as repositories of energy.
As I understand it, for decades the general scientific opinion was that definitive proof of the kinetic theory of gases was still lacking.
As I understand it, the kinetic theory of heat was regarded as plausible, but not proven beyond reasonable doubt.
Einstein's 1905 work in the field of thermodynamics changed that.   Einstein's work in thermodynamics:
- His doctoral thesis on diffusion of sugar molecules in water.
- His paper on brownian motion.
Einstein showed that it was possible to determine Avogadro's number from specific properties of diffusion and brownian motion.
A: One historical answer is Brownian Motion, discovered by botanist Robert Brown and theoretically explained by Albert Einstein to be caused by random motion of particles due to heat:
https://en.m.wikipedia.org/wiki/Brownian_motion
