From existing cosmic ray measurements of theinelastic collision cross sections of nucleons on nuclei of carbon, iron and lead in the range of energies 10² to 10⁴ GeV as well as the measurements of cross sections on air nuclei in the extensive air shower (EAS) regions (10⁵ to 10⁸ GeV), we conclude that the Glauber multiple scattering theory is adequate to account for the data. Recent suggestion of Maor and Nussinov to parametrize the nucleon-nucleon total cross section with a component growing proportional to ln²E (E is the incident energy) is at variance with the EAS data. However the data are consistent with a nucleon-nucleon total cross section rising no faster than lnE in these energy regions.