[en] The low momentum K⁺ is the weakest of the available strongly interacting particles. It has a mean bee path in nuclear matter of about 6 fm which makes it a good probe for studying properties of the nuclear interior. It allows one to build a good microscopic optical potential which can be used to calculate K⁺ nucleus elastic and total cross sections. In the latter case the calculated ratio R_T=[σ_tot(K⁺ A)/A]/[σ_tot(K⁺ d)/2] can be expected to be more reliable because some uncertainties in K⁺N phase shifts will cancel. This ratio can also be measured more reliably than the total cross sections themselves because of cancellation of some systematic errors. We measured the total cross sections of K⁺ on D, ⁶Li, ¹²C, ²⁸Si and ⁴⁰Ca from 488 to 714 MeV/c. The emphasis was placed on extracting values of R_T with a precision of better than 2 percent. The total cross section ratios are found to lie significantly above those predicted by optical potential calculations with 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. (author) 31 refs., 27 figs., 21 tabs.,