What is a "Brass Monkey"?
The story goes that cannonballs used to be stored aboard ship in piles, on a brass frame or tray called a 'monkey'. In very cold weather the brass would contract, spilling the cannonballs: hence very cold weather is 'cold enough to freeze the balls off a brass monkey'.
The phrase is actually first recorded as 'freeze the tail off a brass monkey'.
It therefore seems most likely that the phrase is simply a humorous reference to the fact that metal figures will become very cold to the touch in cold weather.
More Info:
www.oxforddictionaries.com
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