>> In June 1776, Thomas Jefferson wrote the phrase "UNITED STATES OF AMERICA"

In 1507, the German cartographer Martin Waldseemüller produced a world map on which he named the lands of the Western Hemisphere "America" after the Italian explorer and cartographer Amerigo Vespucci). The first documentary evidence of the phrase "United States of America" is from a letter dated January 2, 1776, written by Stephen Moylan, Esq., George Washington's aide-de-camp and Muster-Master General of the Continental Army. Addressed to Lt. Col. Joseph Reed, Moylan expressed his wish to carry the "full and ample powers of the United States of America" to Spain to assist in the revolutionary war effort. 

The first known publication of the phrase "United States of America" was in an anonymous essay in The Virginia Gazette newspaper in Williamsburg, Virginia, on April 6, 1776. The second draft of the Articles of Confederation, prepared by John Dickinson and completed by June 17, 1776 at the latest, declared "The name of this Confederation shall be the 'United States of America.'" The final version of the Articles sent to the states for ratification in late 1777 contains the sentence "The Stile of this Confederacy shall be 'The United States of America'". 

The Declaration of Independence: the Committee of Five presenting their draft to the Second Continental Congress in 1776

In June 1776, Thomas Jefferson wrote the phrase "UNITED STATES OF AMERICA" in all capitalized letters in the headline of his "original Rough draught" of the Declaration of Independence. This draft of the document did not surface until June 21, 1776 and it is unclear whether it was written before or after Dickinson used the term in his June 17 draft of the Articles of Confederation. In the final Fourth of July version of the Declaration, the title was changed to read, "The unanimous Declaration of the thirteen united States of America". The preamble of the Constitution states "...establish this Constitution for the United States of America."

The short form "United States" is also standard. Other common forms are the "U.S.", the "USA", and "America". Colloquial names are the "U.S. of A." and, internationally, the "States". "Columbia", a name popular in poetry and songs of the late 1700s, derives its origin from Christopher Columbus; it appears in the name "District of Columbia". In non-English languages, the name is frequently the translation of either the "United States" or "United States of America", and colloquially as "America". In addition, an abbreviation (e.g. USA) is sometimes used. 

The phrase "United States" was originally plural, a description of a collection of independent states—e.g., "the United States are"—including in the Thirteenth Amendment to the United States Constitution, ratified in 1865. The singular form—e.g., "the United States is"— became popular after the end of the American Civil War. The singular form is now standard; the plural form is retained in the idiom "these United States". The difference is more significant than usage; it is a difference between a collection of states and a unit. 

A citizen of the United States is an "American". "United States", "American" and "U.S." refer to the country adjectivally ("American values", "U.S. forces"). "American" rarely refers to subjects not connected with the United States.
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