  (I) We prove that the (maximum) number of monotone paths in a geometric
triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an
earlier upper bound of $O(1.8393^n)$; the current best lower bound is
$\Omega(1.7003^n)$.
  (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the
number of monotone paths in $G$ can be computed in $O(n^2)$ time.
