Two nights ago, I was lying in bed, thinking about how one might derive the critical buckling load of a column by considering such a column loaded instead like a beam (causing a similar deflected shape). The two equations governing the mid-span moment and deflection of such a beam are, of course, ingrained in my memory (M = wL²/8 and Δ = 5wL⁴/[384EI]), so it only took a moment to figure out the axial load that would reproduce such a state of equilibrium, simulating the condition of buckling. Since the deflection of a buckled column causes a moment equal to the applied load times the deflection, the critical buckling load must be approximately equal to the beam moment divided by the deflection, i.e., P_cr = wL²/8 divided by 5wL⁴/[384EI]) = 9.6EI / L². This is remarkably close to Euler’s famous equation: P_cr = π²EI / L². Anyway, if you want more details, see my short paper on the subject.