The total cross sections for K+ on deuterium, Li-6, C-12, Si-28, and C
a-40 have been measured at four-momenta in the range 480-714 MeV/c. Th
e experimental technique was of the standard transmission type. Becaus
e of its long mean free path in nuclear matter at low energies the Kcan be expected to probe the entire volume of the nucleus. Indeed we f
ound that, to better than 10%, sigma(K+A) = Asigma(K+N) up to A = 40.
This feature was used to study more delicate effects of the nuclear me
dium. Special emphasis was placed on measuring the total cross section
ratios of K+A to K+D with a precision of < 2%. This ratio is a good m
easure of nuclear medium effects because many experimental as well as
theoretical systematic uncertainties cancel in the ratio. We discuss t
he medium dependence as a function of A. The data do not agree with ca
lculations which include only conventional medium effects. This points
towards the need of unconventional nuclear medium effects such as par
tial quark deconfinement, mass rescaling of nuclear pionic fields, or
pion excess in nuclei. Thus far only calculations based on unconventio
nal medium effects for C-12 exist. Calculations for the other nuclei a
re needed before quantitative conclusions can be drawn.