  We summarize significant classical results on (in)determinacy of measures in
terms of their finite positive integer order moments. Well-known is the role of
the smallest eigenvalues of Hankel matrices, starting from Hamburger's results
a century ago and ending with the great progress made only in recent times by
C. Berg and collaborators. We describe here known results containing necessary
and sufficient conditions for moment (in)determinacy in both Hamburger and
Stieltjes moment problems. In our exposition we follow an approach different
from that commonly used. There are novelties well complementing the existing
theory. Among them are: (a) to emphasize on the geometric interpretation of the
indeterminacy conditions; (b) exploit fine properties of the eigenvalues of
perturbed symmetric matrices allowing to derive new lower bounds for the
smallest eigenvalues of Hankel matrices; these bounds are used for concluding
indeterminacy; (c) provide new arguments to confirm classical results; (d) give
new numerical illustrations involving commonly used probability distributions.
