The quality of being fine; delicacy, excellence, or the degree to which something is refined or made thin.
From 'fine' plus '-ness,' the Old English suffix for creating abstract nouns. This is the most common abstract noun form derived from 'fine' and has been used since Old English times.
The word 'fineness' is wonderfully vague—it can mean thinness, beauty, quality, purity, or excellence, depending on context, showing how one simple word can hold multiple precise meanings for speakers who understand the context.
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