  For uniformly chosen random $\alpha \in [0,1]$, it is known the probability
the $n^{\rm th}$ digit of the continued-fraction expansion, $[\alpha]_n$
converges to the Gauss-Kuzmin distribution $\mathbb{P}([\alpha]_n = k) \approx
\log_2 (1 + 1/ k(k+2))$ as $n \to \infty$. In this paper, we show the continued
fraction digits of $\sqrt{d}$, which are eventually periodic, also converge to
the Gauss-Kuzmin distribution as $d \to \infty$ with bounded class number,
$h(d)$. The proof uses properties of the geodesic flow in the unit tangent
bundle of the modular surface, $T^1(\text{SL}_2 \mathbb{Z}\backslash
\mathbb{H})$.
