  Let $\Gamma\backslash\mathbb{H}$ be a Shimura curve, where $\Gamma$
corresponds to the group of units of norm $1$ in an Eichler order $\mathrm{O}$
of an indefinite quaternion algebra over $\mathbb{Q}$. Closed geodesics on
$\Gamma\backslash\mathbb{H}$ correspond to optimal embeddings of real quadratic
orders into $\mathrm{O}$. The intersection numbers of pairs of these closed
geodesics conjecturally relates to the work of Darmon and Vonk on a real
quadratic analogue to $j(\tau_1)-j(\tau_2)$. In this paper, we study the total
intersection number over all embeddings of a given pair of discriminants $D_1,
D_2$. We precisely describe the arithmetic of each intersection, and produce a
formula for the total intersection of $D_1, D_2$. This formula is a real
quadratic analogue of the work of Gross and Zagier on factorizing
$\text{nrd}(j(\tau_1)-j(\tau_2))$. The results are fairly general, allowing for
a large class of non-maximal Eichler orders, and non-fundamental/non-coprime
discriminants. The paper ends with some explicit examples illustrating the
results of the paper.
