He was educated at Beverley grammar school, and was admitted at Trinity Hall, Cambridge on 5 May 1800, becoming a scholar in December 1803.
On 28 January 1814 he was admitted Regius Professor of Civil Law at Cambridge, on the nomination of Lord Liverpool.
[1][3] After the death of his father, and on his own presentation, Geldart became rector of Kirk Deighton in January 1840, and held the benefice for the rest of his life.
[1] Geldart produced an edition in 1836 of An Analysis of the Civil Law by Samuel Hallifax, the set text of his lectures.
[1] Geldart married, on 4 August 1836, Mary Rachel, daughter of William Desborough of Hemingford Grey, Huntingdonshire, who survived him.