The sea-level vertical muon differential momentum spectrum has been measured using the Durham spectrograph MARS in the region 20 GeV/c to 500 GeV/c. The instrumental biases have been studied in detail and allowances made for the particle detector inefficiencies to render to measurement absolute. A simple muon production and propagation model has been used to predict the pion and kaon production spectra from the muon spectrum measurements. It has been found impossible to fit, with any degree of significance, a constant exponent power law pion and kaon production spectrum, having a reasonable value of the K/π ratio (0.15). A better fit is obtained if the exponent is allowed to increase with momentum, and in particular a model with two values of the exponent has been fitted. The muon spectrum has been extrapolated both above and below 500 GeV/c and 20 CeV/c respectively, and at low momenta good agreement is found with the recent "form fit" of De et al. (1972). The present results are compared with previous and contempary measurements of the muon spectrum with the conclusion that there is no evidence, from other recent measurements, that they are incorrect. Comparison with surveys of indirect measurements at higher energies however suggest that the muon spectrum cannot continue in this enhanced fashion much beyond 1000 GeV. Finally an absolute integral rate experiment has been performed using MARS as a range spectrograph, and the intensity above 7.12 GeV/c is found to be in agreement with a previous similar measurement made with the instrument. Further it is concluded that the intensity at this momentum is in agreement with the extrapolation of the differential spectrum measurements below 20 GeV/c
