  We investigate the stability of a one-parameter family of periodic solutions
of the four-vortex problem known as `leapfrogging' orbits. These solutions,
which consist of two pairs of identical yet oppositely-signed vortices, were
known to W.\ Gr\"obli (1877) and A.\ E.\ H.\ Love (1883), and can be
parameterized by a dimensionless parameter $\alpha$ related to the geometry of
the initial configuration. Simulations by Acheson (2000) and numerical Floquet
analysis by Toph{\o}j and Aref (2012) both indicate, to many digits, that the
bifurcation occurs when $1/\alpha=\phi^2$, where $\phi$ is the golden ratio.
This study aims to explain the origin of this remarkable value. Using a trick
from the gravitational two-body problem, we change variables to render the
Floquet problem in an explicit form that is more amenable to analysis. We then
implement G. W. Hill's method of harmonic balance to high order using computer
algebra to construct a rapidly-converging sequence of asymptotic approximations
to the bifurcation value, confirming the value found earlier.
