  Dirac's approach to gauge symmetries is discussed. We follow closely the
steps that led him from his conjecture concerning the generators of gauge
transformations {\it at a given time} --to be contrasted with the common view
of gauge transformations as maps from solutions of the equations of motion into
other solutions-- to his decision to artificially modify the dynamics,
substituting the extended Hamiltonian (including all first-class constraints)
for the total Hamiltonian (including only the primary first-class constraints).
We show in detail that Dirac's analysis was incomplete and, in completing it,
we prove that the fulfilment of Dirac's conjecture --in the "non-pathological"
cases-- does not imply any need to modify the dynamics. We give a couple of
simple but significant examples.
