The total cross sections for K+ mesons on carbon and deuterium nuclei have been measured at eleven momenta in the range 450740 MeV/c. The experimental technique was of the standard transmission type. The K+ meson is the least strongly interacting of available hadronic probes, with a long mean free path in nuclear matter. At low incident momentum the K+N interaction is dominated by the S11 phase shift and varies slowly with energy. These characteristics make the K+ an ideal tool for probing the nuclear volume to reveal nuclear medium effects. Measurements of the ratio of the total cross sections, per nucleon, of K+-12C to K+-d have been suggested as a way to reveal effects of the nuclear medium. The total cross section ratios are found to lie significantly above those predicted by the usual nuclear medium corrections. This suggests that novel phenomena are taking place within the nucleus. Several models which incorporate such phenomena are discussed, including nucleon swelling, mass rescaling, nuclear pions, and relativistic effects.