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One of the most prominent and well established features in the primary energy spectrum of cosmic rays observed at the Earth is the change of slope occurring at several 10(^15) eV. A comprehensive survey of experimental EAS data is used to establish the integral size spectrum for electrons and muons at sea level and for electrons at mountain altitudes. A model for the diffusion of protons in the Galaxy is developed based on experimental observations of the structure of hydrogen in the interstellar medium and of the magnetic field strengths and their orientations in galactic space. A numerical treatment of the diffusion problem is adopted. A pre-diction is made of the primary cosmic ray proton energy spectrum at the top of the atmosphere. A survey of the data on cosmic ray primaries with energies below ~10(^11) eV obtained by balloon and satellite experiments, was used to establish the relative abundances of all cosmic rays. By assuming that the cosmic ray composition remains the same at EAS energies the primary spectrum representing diffusion of cosmic rays with mixed composition is derived. Comparison of the theory with experiment was made by converting the integral primary energy spectrum representing diffusion to the integral size spectrum by using results of EAS simulations through the atmosphere. Excellent agreement of theory with experiment is obtained provided that the primary cosmic radiation is protonic. The problems encountered with ensuring isotropy of cosmic rays at the Earth are dealt with in great detail and are shown to create little problem over the part of the spectrum considered provided the solar system lies in the centre of the spiral arm. Finally, a model is proposed as an alternative to that of diffusion which provides a qualitative but adequate explanation for the origin of the 'knee'.