The onset of buoyancy-driven convection in an initially motionless isothermal fluid layer is analyzed numerically. The infinite horizontal fluid layer is suddenly cooled from above to relatively low temperature. The rigid lower boundary remains at the initial temperature. In the present transient system, when the Rayleigh number Ra exceeds 1101, thermal convection sets in due to buoyancy force. To trace the temporal growth rates of the mean temperature and its fluctuations we solve the Boussinesq equation by using the finite volume method. We suggest that the system begins to be unstable when the growth rate of temperature disturbances becomes equal to that of the conduction field. Three different characteristic times are classified to interpret numerical results clearly: the onset time of intrinsic instability, the detection time of manifest convection and the undershoot time in a plot of the cooling rate versus time. The present scenario is that the thermal instability sets in at the critical time, then grows super-exponentially up to near the undershoot time, and between these two times the first visible motion is detected. Numerical results are compared with available experimental data. It is found that the above scenario looks promising and the critical time increases with decreasing the Prandtl number Pr and also the Rayleigh number Ra.