A new characterization method, "Macroscopic Composition Gradient (MCG) Method" is proposed to investigate the phase transformations near the phase boundaries, such as the solubility limit, order/disorder line and so on. Since the macroscopic composition gradient in the alloy is prepared so as to step over the phase boundary, the morphological transition of critical phenomena at the phase boundary can be observed by means of analytical transmission electron microscopy. By utilizing this method, the critical minimum size of stable precipitate in the vicinity of edge of miscibility gap is experimentally determined for the Ni3Si in Ni-Si, Ni3Al in Ni-Al, Cu4Ti in Cu-Ti and Co in Cu-Co binary alloy systems. The results are as follows: The critical nucleus size shows a steep increase up to several tens of nm in a very narrow composition range less than 0.3at% from the phase boundary. The Gibbs-Thomson relation and the conventional nucleation theory statistically rationalize such the composition dependence of nucleus size change. However, the nucleus formation is kinetically never rationalized by the conventional nucleation theories. The phase decomposition of supersaturated solid solution progresses by a mechanism of spinodal phase decomposition, even in the composition range near the solubility limit, i.e. a so-called Nucleation- Growth region. Such the phase decomposition behavior is never rationalized by the Boltzmann- Gibbs free energy, which is based on the extensive entropy. The experimental facts obtained here are explained by Tsallis's non-extensive entropy. It should be noted that the present experiments can systematically be conducted in the composition range very near the solubility limit where they has hardly been examined in the past. The MCG method proposed here is considered to open a new way to investigate the microstructure evaluation, particularly for the critical phenomena near the phase boundary.