Educated at Charterhouse and Trinity College, Cambridge,[2] he entered the Middle Temple in 1838, practising as a special pleader in 1845 and being called in 1846.

Joining the midland circuit, he acquired a good practice, and on the recommendation of Chief Justice Sir William Erle he was appointed as the first Chief Justice of the Supreme Court of Queensland in Queensland, Australia on 21 February 1863; he served until his retirement on 24 June 1879.

For instance he invented the number systems of tessarines and coquaternions, and worked with Arthur Cayley (1821–1895) on the theory of linear algebra.

Like many young mathematicians he attacked the problem of solving the quintic equation, notwithstanding Abel–Ruffini theorem that a solution by radicals was impossible.

In this field Cockle achieved some notable results, amongst which is his reproduction of Sir William R. Hamilton's modification of Abel's theorem.