  We present a discussion on how to define the running electromagnetic coupling
constant at $M_Z$ or some other intermediate scale as e.g. $m_\Upsilon$. We
argue that a natural definition consistent with general requirements of the
renormalization group should be based on Euclidean values of the momentum of
the photon propagator as the appropriate scale. We demonstrate in an explicit
example of evaluating the running coupling constant at the scale of the
$\Upsilon$ resonance mass that the usual definition of the hadronic
contribution with a principal value prescription is inconsistent. In the
determination of the value of $\alpha$ at $M_Z$ the numerical difference due to
using a Euclidean definition rather than the principal value one is comparable
in size to the errors caused by existing experimental and QCD inputs to the
evaluation of $\alpha(M_Z)$.
