  This paper generalizes Manin's approach towards a geometrical interpretation
of Arakelov theory at infinity to linear cycles on projective spaces. We show
how to interpret certain non-Archimedean Arakelov intersection numbers of
linear cycles on $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits
building associated to PGL(n). This geometric setting has an Archimedean
analogue, namely the Riemannian symmetric space associated to SL(n,C), which we
use to interpret analogous Archimedean intersection numbers of linear cycles in
a similar way.
