  Accurate fits to $pp$ and $\bar pp$ cross section data up to Tevatron
energies, incorporating the constraints imposed by analyticity and unitarity,
successfully predict the results of recent LHC and cosmic ray measurements, and
suggest that the cross sections approach a black disc limit asymptotically. The
approach to the limit is, however, very slow. We present a simple geometric
picture which explains these features in a natural way. A black disc of
logarithmically growing radius is supplemented by a soft `edge' whose
properties are invariant with energy. The constancy of the edge results in the
prediction that the quantity $(\sigma^{TOT}-2\sigma^{El})/\surd\sigma^{TOT}$
approaches a constant at high energy. Using the existing fits, this prediction
appears to be verified. The value of the limiting constant allows an estimate
of the thickness of the edge, which turns out to be on the order of $1\,{\rm
fm}$. One thus arrives at a picture where the proton-proton scattering at lower
energies is dominated by what becomes the edge, while at higher energies it is
dominated by the disc. The crossover between the two regimes is only at $\surd
s\geq $ 10 TeV, accounting for the slow approach to asymptotic behavior. Some
questions as to the nature of the edge are discussed.
