How did Lorentz find his theorem of corresponding states?

I offer a reconstruction of how Lorentz found his theorem of corresponding states or, in modern terms, the Lorentz invariance of the source-free Maxwell equations. I show that, to arrive at Lorentz's theorem, both the approximate version of 1895 and the exact version of 1899/1904, Lorentz had to do little more than regroup terms and repeatedly apply the chain rule of differentiation. My reconstruction should thus help dispel the aura of complexity that still surrounds the theorem. I also show how the theorem can be extended to Maxwell's equations with sources and briefly review how Lorentz, after 1905, went from considering Lorentz-transformed quantities purely as auxiliary variables to recognizing them as the measured quantities for moving observers.