  Geometrical interpretation on U(1) gage theory of Dirac monopole, introduced
here from the line integral\cite{Brandt} form of vector potentials, shows the
gauge representation be multi-valued. In this paper, we construct Euclidean
form of U(1) gauge theory on Dirac monopole based on the geometrical
interpretation, derive the Dirac quantization condition geometrically. Even
several theoretical methods which leads to Dirac quantization $eg=l/2$ has been
developed, these are only possible under the postulate that gauge
representation is single-valued. Using the line integration representation of
vector potential, we derived the gauge equation which satisfies under the
`phase gauge transformation' ($\mathbf{A}\to\mathbf{A}'=\mathbf{A}, \
\psi\to\psi'=e^{ie\Lambda}\psi$), also showed that the Dirac quantization
condition can be derived by the geometrical boundary condition on this gauge
equation, without any supposing of canonical quantization condition. We finally
explain geometrical meaning of constant phase of gauge and how the Dirac
quantization holds single valued gauge representation.
