  We make a detailed eikonal fit to current data on the total and elastic
scattering cross sections, the ratios $\rho$ of the real to the imaginary parts
of the forward elastic scattering amplitudes, and the logarithmic slopes $B$ of
the differential cross sections $d\sigma/dt$ at $t=0$, for proton-proton and
antiproton-proton scattering at center-of-mass energies $W$ from 5 GeV to 57
TeV. The fit allows us to investigate the structure of the eikonal amplitudes
in detail, including the impact-parameter structure of the energy-independent
edge in the scattering amplitude shown to exist by Block {\em et al.}
\cite{edge}. We show that the edge region has an essentially fixed shape with a
peak at approximately the "black disk" radius $R_{\rm tot}=\sqrt{\sigma_{\rm
tot}/2\pi}$ of the scattering amplitude, a constant width $t_{\rm edge}\approx
1$ fm, and migrates to larger impact parameters with increasing energy
proportionally to $R_{\rm tot}$. We comment on possible physical mechanisms
which could lead to the edge. We show that the eikonal results for the cross
sections and $\rho$ values are described to high accuracy by analytic
expressions of the forms used in earlier analyses by Block and Halzen, and
extend the result to the elastic-scattering slope parameter $B$. These
expressions provide simple extrapolations of the results to much higher
energies. Finally, we calculate the survival probabilities for large rapidity
gaps in the scattering.
