On Gauss' characterization of the normal distribution

Consider the following problem: if the maximum likelihood estimate of a location parameter of a population is given by the sample mean, is it true that the distribution is of normal type? The answer is positive and the proof has been given by Gauss (1809) although without using the likelihood terminology. We revisit this result in a modern context and present a simple and rigorous proof. Extensions to a $p$-dimensional population and to the case with a parameter additional to that of location are also considered