How did American currency come to be called dollars and cents?

Dollar was the English spelling of the German Taler (a silver coin first issued in 1519).

American colonists used the word dollar to describe the Spanish peso circulating from South America, and when it came time to devise a system of currency (in 1792), the United States government adopted the dollar as its basic monetary unit.

The word cent meant one-hundredth of a dollar, following the decimal system of coinage first proposed by Gouverneur Morris (1752-1816).

Morris was a New York–born statesman who served as assistant to the superintendent of finance under the Articles of Confederation, from 1781 to 1785.

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