He was born in Dublin, Ireland, the fourth child of John Smith, a barrister. He was educated until he was fifteen by his mother and private tutors, before attending Rugby school from 1841. He suffered from poor health, missing his final year at Rugby while convalescing in Italy. In late 1844 he tried for and obtained the Balliol scholarship. While at Oxford his health did not improve, he was struck down with malaria while in Frascati in 1845 and he did not return to Oxford until 1847. He graduated in 1849 with a double first in mathematics and classics. Smith remained at Balliol, becoming a fellow (1850), then working as a tutor before being appointed Savilian Professor of Geometry in 1861. He was elected to the Royal Society and to the Royal Astronomical Society in 1861.
His main published work was his Report on the Theory of Numbers (in five parts 1859-65).