  We investigate the extrinsic geometry of causal sets in $(1+1)$-dimensional
Minkowski spacetime. The properties of boundaries in an embedding space can be
used not only to measure observables, but also to supplement the discrete
action in the partition function via discretized Gibbons-Hawking-York boundary
terms. We define several ways to represent a causal set using overlapping
subsets, which then allows us to distinguish between null and non-null bounding
hypersurfaces in an embedding space. We discuss algorithms to differentiate
between different types of regions, consider when these distinctions are
possible, and then apply the algorithms to several spacetime regions. Numerical
results indicate the volumes of timelike boundaries can be measured to within
$0.5\%$ accuracy for flat boundaries and within $10\%$ accuracy for highly
curved boundaries for medium-sized causal sets with $N=2^{14}$ spacetime
elements.
