  Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a
partition into circuits. They defined the class of ``finite matching
extendable'' matroids and showed in their compactness theorem that those
matroids always admit such a partition. Their proof is based on Shelah's
singular compactness technique and a careful analysis of certain
$\triangle$-systems.
  We provide a short, simple proof of their theorem. Then we show that a
finitary binary oriented matroid can be partitioned into directed circuits if
and only if, in every cocircuit, the cardinality of the negative and positive
edges are the same. This generalizes a former conjecture of Thomassen, settled
affirmatively by the second author, about partitioning the edges of an infinite
directed graph into directed cycles. As side results, a Laviolette theorem for
finitary matroids and a Farkas lemma for finitary binary oriented matroids are
proven. An example is given to show that, in contrast to finite oriented
matroids, `binary' cannot be omitted in the latter result.
