I was reading Thermodynamics by Enrico Fermi first they give this function ##\frac{Q_2}{Q_1}=f(t_2,t_1)=\frac{g(t_2)}{g(t_1)}##, where ##g## is monotone increasing function. Idea is if we have scales of temperature ##r,s## (monotone increasing), but and their inverses is also scales of empirical temperatures ##r^{-1},s^{-1}##. ##f(t_2,t_1)=f(s(t_2),s(t_1))##, because function ##f##, doesn't depend of scales, and depend only of temperatures (Carnot's theorem). If we has two objective temperature scales they must has homomorphism between scales, and they must be linear, because temperature is interval data e.g.
##s(t)=\alpha x+\beta, \alpha>0, \beta## is some parameter. I must pack properties that homomorphism is linear function between ##s## and ##r##. Let it be ##s(t_2-t_1)=s(t_2)-s(t_1)##
We have ##f(t_2,t_1)=f(t_2-t_1,0)=f(s(t_2-t_1),s(0))=f(s(t_2-t_1)-s(0),0)=f(s(t_2),s(t_1))=f(s(t_2)-s(t_1),0)\implies
s(t_2-t_1)-s(0)=s(t_2)-s(t_1)##, set ##t_2=t_1##, this is Caushy equation and this solution is ##s(t)=ct##. As I says
scale is and function ##F(t)=r(s^{-1}(t)),r(t_2-t_1)=r(t_2)-r(t_1), s(t_1)=x, s(t_2)=y\implies F(x-y)=F(x)-F(y) \implies F(x)=\alpha x+\beta##(Pfanzagl, Theory of Measurement, page 98)
