  An analogue of the Euclidean algorithm for square matrices of size 2 with
integral non-negative entries and strictly positive determinant $n$ defines a
finite set $\mathcal{R}(n)$ of Euclid-reduced matrices corresponding to
elements of $\{(a, b, c, d) \in \mathbb{N}^4 | n = ab - cd,\ 0 \le c, d < a,
b\}$. With Popeye's help[2] on the use of sails of lattices we show that
$\mathcal{R}(n)$ contains $\sum{d|n, d^2 \ge n} (d + 1 - n/d)$ elements.
