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Born in South Carolina on May 22, 1783, Crenshaw was the first graduate of the South Carolina College at Columbia, later renamed the University of South Carolina. He became active in politics and was elected to the South Carolina legislature in 1812.
Several years later in 1819 he moved to Cahawba, the first state capital of Alabama. There, he was appointed a judge of the circuit court from 1821–1838, of the state supreme court from 1832, and as chancellor of the southern division of the state's courts. He died in 1847.
In 1866 the newly created Crenshaw County, Alabama was named in his honor.