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A reformulation of the optical model is developed in which the real parts of the potential are obtained from nuclear-matter distributions and the nucleon-nucleon force. The model is applied to proton elastic scattering data at 14.5, 30.3, and 40.0 MeV and succeeds in fitting the data as well as, or better than, the standard optical model despite the fact that two fewer parameters are needed in the new model. Values, accurate to a few percent, are obtained for the nuclear rms matter radii which are independent of the incident proton energy. These values are greater than the corresponding rms proton radii obtained from electron scattering and muonic x-ray work, and indicate that nuclear neutron rms radii are greater than nuclear proton rms radii by about 0.6 F. Information is also obtained concerning the spin-isospin-independent part of the nucleon-nucleon force, indicating a mean-square radius of 2.25\ifmmode\pm\else\textpm\fi{}0.6 ${\mathrm{F}}^{2}$ and a volume integral of 400\ifmmode\pm\else\textpm\fi{}20 MeV ${\mathrm{F}}^{3}$. The neutron and proton density distributions found from this work and muonic studies are used to calculate the imaginary potential, and this is compared with the phenomenological form found in the analyses performed with the new model. The good measure of agreement between the two potentials indicates that the model can be extended to include this term in a more logical manner, and with fewer parameters, than in the standard formulation of the optical model. The model is readily extended, in appropriate cases, for use with complex particles.