The Collaborative International Dictionary of English v.0.48:

Fineness \Fine"ness\, n. [From Fine, a.]
   1. The quality or condition of being fine.
      [1913 Webster]

   2. Freedom from foreign matter or alloy; clearness; purity;
      as, the fineness of liquor.
      [1913 Webster]

            The fineness of the gold, and chargeful fashion.
                                                  --Shak.
      [1913 Webster]

   3. The proportion of pure silver or gold in jewelry, bullion,
      or coins.
      [1913 Webster]

   Note: The fineness of United States coin is nine tenths, that
         of English gold coin is eleven twelfths, and that of
         English silver coin is [frac925x1000].
         [1913 Webster]

   4. Keenness or sharpness; as, the fineness of a needle's
      point, or of the edge of a blade.
      [1913 Webster]

WordNet (r) 3.0 (2006):

fineness
    n 1: the quality of being very good indeed; "the inn is
         distinguished by the fineness of its cuisine" [syn:
         fineness, choiceness]
    2: the property of being very narrow or thin; "he marvelled at
       the fineness of her hair" [syn: fineness, thinness]
    3: having a very fine texture; "the fineness of the sand on the
       beach" [syn: fineness, powderiness]
    4: the quality of being beautiful and delicate in appearance;
       "the daintiness of her touch"; "the fineness of her features"
       [syn: daintiness, delicacy, fineness]