thermodynamics - Was Joule's experiment able to show: thermal energy = $mgh$

Is my understanding of this experiment correct? There is the quantity $MC\delta \theta$ associated to the heat which I think is not an absolute value to be compared with $mgh$ since the $\delta \theta$ could be defined arbitrarily. So this experiment does not prove : thermal energy = $mgh$ it could however show a linear relation between $\delta \theta$ and $h$.