The doubly indexed Whitney numbers of a finite, ranked partially ordered set
L
L
are (the first kind)
w
i
j
=
∑
{
μ
(
x
i
,
x
j
)
:
x
i
,
x
j
∈
L
{w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L}
with ranks
i
,
j
}
i,j\}
and (the second kind)
W
i
j
=
{W_{ij}} =
the number of
(
x
i
,
x
j
)
({x^i},{x^j})
with
x
i
⩽
x
j
{x^i} \leqslant {x^j}
. When
L
L
has a
0
0
element, the ordinary (simply indexed) Whitney numbers are
w
j
=
w
0
j
{w_j} = {w_{0j}}
and
W
j
=
W
0
j
=
W
j
j
{W_j} = {W_{0j}} = {W_{jj}}
. Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of
k
k
-dimensional faces for any
k
k
, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope
P
P
inside the visible boundary as seen from a distant point on a generating line of
P
P
. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley’s theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly
q
q
sources (generalizing Rényi’s enumeration of permutations with
q
q
"outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph’s having a unique source.