Colgate University is a private college in Hamilton, New York. The liberal arts college was founded in 1819 as the Baptist Education Society of the State of New York and operated under that name until 1823, when it was renamed Hamilton Theological and Literary Institution, often called Hamilton College, then Madison College, and its present name since …Colgate University is a private college in Hamilton, New York. The liberal arts college was founded in 1819 as the Baptist Education Society of the State of New York and operated under that name until 1823, when it was renamed Hamilton Theological and Literary Institution, often called Hamilton College, then Madison College, and its present name since 1890.