  We use a dictionary between lattice point counting inside dilated
d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed
horosphere that intersect a ball of increasing radius, to obtain two types of
results. Firstly, via an $L^2$-integral error estimate for Euclidean counting,
we prove effective equidistribution results for a family of expanding
horospheres in the locally symmetric space SO(d) \ SL(d,R) / SL(d,Z). Secondly,
we derive from uniform error estimates in Euclidean counting, error terms for
counting SL(d,Z)-orbit points in a certain increasing family of subsets in
SO(d) \ SL(d,R) (which we call truncated chimneys), and for counting the number
of lifts of a closed horosphere that intersect a ball with large radius.
