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The Collaborative International Dictionary
Leech line

Leech \Leech\, n. [Cf. LG. leik, Icel. l[=i]k, Sw. lik boltrope, st[*a]ende liken the leeches.] (Naut.) The border or edge at the side of a sail. [Written also leach.]

Leech line, a line attached to the leech ropes of sails, passing up through blocks on the yards, to haul the leeches by.
--Totten.

Leech rope, that part of the boltrope to which the side of a sail is sewed.

Wiktionary
leech line

alt. (context nautical English) A line for tightening the leech of a triangular mainsail to prevent it from fluttering. n. (context nautical English) A line for tightening the leech of a triangular mainsail to prevent it from fluttering.