The Collaborative International Dictionary of English v.0.48:

Fine \Fine\ (f[imac]n), v. i.
   To become fine (in any one of various senses); as, the ale
   will fine; the weather fined.

   To fine away, down, off, gradually to become fine; to
      diminish; to dwindle.

            I watched her [the ship] . . . gradually fining down
            in the westward until I lost of her hull. --W. C.
                                                  Russel.
      [Webster 1913 Suppl.]