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After studying medicine, Swan began to practice in Bound Brook, from 1800-1806 and then moved to Somerville from 1806 to 1809. He was elected to U.S. House of Representatives for the Seventeenth, Eighteenth, Nineteenth, Twentieth, and Twenty-first United States Congresses, serving from March 4, 1821 to March 3, 1831. He was an at-large representative for all but his second term, during which he represented the 2nd congressional district. He retired without seeking renomination. Although he did not formally represent a party when elected to the House, he eventually was affiliated with the Whig Party.
After his congressional career, he returned to practicing medicine.
He is buried in the Presbyterian Cemetery in Bound Brook.