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The motto was picked up during the late 17th and early 18th century as one of several that revolutionaries rallied around. Liberty represents freedom, the French Revolution was an opposition of the monarchy and was largely spurred on by the American Revolution, which predated it by over a decade. Equality represents just that. Fraternity means brotherhood. The French Revolution was inspired by the same ideas of inalienable rights, popular sovereignty, etc that the US Constitution is based on.

Also, Liberty was a popular character in French political movements. In fact the Statue of Liberty, which the US received from France as a gift, was literally a statue of the character Lady Liberty (who in her own time was iconic as Uncle Sam is to Americans). The motto was adapted into the 1958 constitution and is now ingrained into French history. It is as common to the French as E Pluribus Unum or In God We Trust is to Americans.

French Motto

France Motto

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I think the motto comes from the time after the King was overthrown. Révolution française

Napoleon and his advisors i think