1) A serving of food, as here at a funeral feast.

1586 ‘a dinner to be made on the day of my burial for ten meaces of my neighbours at 2s a meace’, Leeds.

spellings meace
places Leeds
dates 1586

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2) The quantity of milk given by a cow at one milking (OED).

1738 she had milked about a mess of milk in a jugg, Halifax.

places Halifax
dates 1738

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Photo by Kreuzschnabel CC BY-SA 3.0