  Fix a translation surface $X$, and consider the measures on $X$ coming from
averaging the uniform measures on all the saddle connections of length at most
$R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal
to the Lebesgue measure on $X$. We also show that any weak limit of a
subsequence of the counting measures on $S^1$ given by the angles of all saddle
connections of length at most $R_n$, as $R_n\to\infty$, is in the Lebesgue
measure class. The proof of the first result uses the second result, together
with the result of Kerckhoff-Masur-Smillie that the directional flow on a
surface is uniquely ergodic in almost every direction.
