The equation of heat conduction is solved for a horizontally stratified medium initially at constant temperature and subject to a step-function surface temperature change. The problem is solved by Laplace transformation and by applying a concept of further stratifications of the medium into unitary layers of constant ratio of thickness to the square root of thermal diffusivity. The solution takes the form of an infinite sum of complementary error functions, with coefficients given by recursion relations, is suitable for numerical applications and offers an attractive alternative to harmonic and quasi transient approaches in calculating the penetration of transient surface temperature variations into a layered medium. Use of the theorem of superposition yields a general expression for an arbitrary surface temperature function. The utility of the theory is illustrated by modelling examples of palaeoclimatically induced subsurface temperature and heat flow perturbations.