In 1970, on purely theoretical grounds, all total hadronic total cross
sections were predicted to increase without limit for higher and higher
energies. This was contrary to the conventional belief at that time. In 1978,
an accurate phenomenological model was formulated for the case of proton-proton
and antiproton-proton interactions. The parameters for this model were slightly
improved in 1984 using the additional available experimental data. Since then,
for thirty years these parameters have not changed. This development, including
especially the difficult task of formulating this phenomenological model and
the comparison of the predictions of this model with later experimental
results, is summarized.Comment: 14 pages, 6 figures, Contribution to the special issue of the
International Journal of Modern Physics A on "Elastic and diffractive
scattering" coordinated by Christophe Royo
