The purpose of this paper is to find cosmological evidence in support of Dirac's hypothesis of a variable gravitational number. A theoretical apparent magnitude vs. red shift relationship is constructed for galaxies under the effects of a variable G (Brans-Dicke) cosmology. A time-dependent gravitational number has the effect of increasing the luminosity of a galaxy and of reducing its age (as compared to a galaxy under constant G conditions). The construction of a theoretical apparent magnitude-red shift relationship requires a knowledge of the luminosity-history of a galaxy. Theoretical evolutionary histories have been computed by Beatrice M. Tinsley (1968). Adopting a set of initial conditions and a simple stellar birth rate function, she evolved a galaxy in one billion year intervals from its formation to its present age. Using theoretical stellar evolutionary tracks she was able to construct a luminosity-age relationship for every galactic type. Using Tinsley's results, a theoretical apparent magnitude-red shift relationship is constructed, and then compared to the empirical apparent magnitude-red shift plot which contains quasi-stellar objects (QSOs) as well as normal galaxies. It is hoped that the QSOs can be explained as earlier phases of a normal galaxy under a larger gravitational number. The results of this paper are inconclusive. A variable G cosmology with w (a dimensionless constant related to the rate at which G changes with time) equal to 2.5, and H (Hubble's constant) equal to 75 km/sec/Mpc, results in a theoretical apparent magnitude-red shift relationship which agrees fairly well with the empirical plot. But the brightness fluctuations and the compactness of the QSOs still remain unexplained. And even more damaging is the fact that if QSOs are to be considered as earlier phases of a normal galaxy, then there should be five QSOs for every normal galaxy. The number of normal galaxies which have been observed, however, exceeds the observed number of QSOs by a factor of 10⁸.