  Euler's celebrated buckling formula gives the critical load $N$ for the
buckling of a slender cylindrical column with radius $B$ and length $L$ as \[ N
/ (\pi^3 B^2) = (E/4)(B/L)^2, \] where $E$ is Young's modulus. Its derivation
relies on the assumptions that linear elasticity applies to this problem, and
that the slenderness $(B/L)$ is an infinitesimal quantity. Here we ask the
following question: What is the first nonlinear correction in the right
hand-side of this equation when terms up to $(B/L)^4$ are kept? To answer this
question, we specialize the exact solution of incremental non-linear elasticity
for the homogeneous compression of a thick compressible cylinder with
lubricated ends to the theory of third-order elasticity. In particular, we
highlight the way second- and third-order constants ---including Poisson's
ratio--- all appear in the coefficient of $(B/L)^4$.
