  We give a classification of quadratic harmonic morphisms between Euclidean
spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence
between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford
systems. In the case $ {\Bbb R}^{4}\longrightarrow {\Bbb R}^{3} $, we determine
all quadratic harmonic morphisms and show that, up to a constant factor, they
are all bi-equivalent (Definition 3.2) to the well-known Hopf construction map
and induce harmonic morphisms bi-equivalent to the Hopf fibration ${\Bbb S}^{3}
\longrightarrow {\Bbb S}^{2}$.
