Something that occurs too early; starting too soon.
This phrase is believed to come from foot racing. There are foot races that begin with the firing of a pistol into the air. The sound of the shot basically means "GO!" for the racers. However, sometimes not all of the runners will wait until they hear the shot. Yes, there are times when a runner will start the race prematurely before the starter even has a chance to fire his pistol, and thus, that runner is jumping the gun. Or in other words, the runner is starting before he's supposed to. This phrase is now used for other things that are done too soon.

The expression goes at least as far back as the year 1830, when the newspaper Jacksonville Journal Courier uses it in an advice column where people would write in to a woman named Abby for counsel:

    "When you describe his condition as 'dying,' you create the impression that you are rushing him to the 
     cemetery. He could live quite a while, so don't jump the gun."

To give some context for the quote above: This was part of Abby's response to someone who said his father-in-law was "dying of liver cancer." The writer's in-laws were upset at his use of the word "dying," and so he wondered if they were being too sensitive, or if he was in the wrong.

Reference: newspaperarchive
* My wife thinks I might be jumping the gun here since it's a bit expensive, but I think I'm going to use the money I've been saving for a few years now to buy this brand new car. - Meanings and Origins of Phrases

Note: The origins for most phrases and popular idioms cannot be said with a certainty. What's provided are theories that may be plausible to how a phrase originated, but not necessarily so. 

In addition, quotes that contain a particular phrase may be taken from old newspapers, poems, or books that were written centuries ago, but this by no means confirms that the phrase originates from said newspapers, poems, or books. In all likelihood, if an expression is being used in a newspaper, it's probably already a well known saying and is from an older time.
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Idioms beginning with the letter - J.
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