He was admitted to Trinity College, Cambridge, in 1812 as a sizar, graduating with a BA in 1816 (5th Wrangler) and MA in 1821.
He was appointed Assistant Professor at the Royal Naval Academy, Portsmouth (a post which he held with the curacy of St George's, Portsea) in 1816, before being appointed in 1822 Senior Professor of Mathematics at Addiscombe College, Croydon, Surrey.
Vibart's detailed history of Addiscombe contains anecdotal material about Cape, commenting that he was "the most remarkable member of the staff of the College during the whole course of its existence ...
[2] Cape was noted for his dry, sarcastic wit, and for his strong Northumbrian accent, as well as being "an excellent teacher and disciplinarian, with a keen sense of humour".
He retired on a very substantial pension, and died – a convivial bachelor – at Croydon on 9 September 1868 aged 75, leaving £12,000.