How did people come up with the formula: $$PV = nRT = k_BT \qquad$$?
I see two possible ways:
1.) We did some measurements and defined $$kB:=\frac{PV}{T},$$ remarking that $\frac{PV}{T}$ is constant.
2.) We had $k_B$ from somewhere else, derived $PV$ must equal $k_BT$. Then we made some measurements and voila it seemed to be true.
3.) Neither of the two is true. If so please explain in detail, what happened.
Remark: Beside of this question I want to understand how people came up with the equipartition theorem. Assuming that we found the following formulas independently: $$(1)\ PV = k_BT \qquad (2)\ PV = \frac{2}{3}K \qquad with\ K\ the\ kinetic\ energy\ of\ the\ gas.$$ One could argue that we get $N\frac{3}{2}k_BT = K$, so there must be $\frac{1}{2}k_BT$ kinetic energy per molecule. And assuming only translation we get the equipartition theorem. Or probably this theory of mine is again false...
I guess I kinda mix up some things a bit, but I think I made it clear what I want to understand.
Remark 2: It is important to me to understand how people came up with these ideas. It is interesting if you note that certain derivations come up with the same result and therefor confirm the idea, but what I actually want to know is, how those formulas where discovered in the first place.
