We have determined the variations in the emission measure of the solar corona using EUV Imaging Telescope/Solar and Heliospheric Observatory observations of iron lines in a quiet region of the Sun. The emission measure is found to vary significantly in at least 85% of all the pixels within 42 minutes. The variations are interpreted as heating events that bring chromospheric material above the one million degree threshold of the observed lines and that cool the coronal plasma below that limit. A method to assess heating events has been developed. The thermal energy input by such microflares is calculated from the observed increases in emission measure and the derived temperature. Heating events have been found in the range from 8 ? 10 -->24 to 1.6 ? 10 -->26 ergs. The energy input by ?3 ? events of the emission measure increase the amounts to about 16% of the average radiated power of the coronal plasma in the quiet corona. The frequency distribution of microflares is an approximate power law of the form f(E)=f -->0E-?, with a power-law index ? between 2.3 and 2.6. Since the low-energy cutoff is due to sensitivity limitations and the power-law index is steeper than 2, these observations demonstrate the possibility that microflares dominate the energy input into the quiet corona. The observed power law would have to continue to about 3 ? 10 -->23 ergs in order to match the observed minimum heating requirement.