  An orthogonality space is a set together with a symmetric and irreflexive
binary relation. Any linear space equipped with a reflexive and anisotropic
inner product provides an example: the set of one-dimensional subspaces
together with the usual orthogonality relation is an orthogonality space. We
present simple conditions to characterise the orthogonality spaces that arise
in this way from finite-dimensional Hermitian spaces. Moreover, we investigate
the consequences of the hypothesis that an orthogonality space allows gradual
transitions between any pair of its elements. More precisely, given elements
$e$ and $f$, we require a homomorphism from a divisible subgroup of the circle
group to the automorphism group of the orthogonality space to exist such that
one of the automorphisms maps $e$ to $f$, and any of the automorphisms leaves
the elements orthogonal to $e$ and $f$ fixed. We show that our hypothesis leads
us to positive definite quadratic spaces. By adding a certain simplicity
condition, we furthermore find that the field of scalars is Archimedean and
hence a subfield of the reals.
