For six months he was placed at Market Harborough in the school of Stephen Addington, but his early life was mainly passed under his father's roof, and he was self-taught in the rudiments of Greek.
On 21 September 1790 he married Mary, the daughter of David Atchison of Weedon, and in the following July he settled for a short time at Ware in Hertfordshire, but after a few months he moved first to Enfield and then to Cheshunt.
Cogan was elected minister of the chapel in Crossbrook Street, Cheshunt, in 1800, and in January of the following year he was also appointed by the dissenting congregation at Walthamstow.
His portrait in life-size was painted at the cost of his pupils by Thomas Phillips, R.A., and engraved by Samuel Cousins, and the picture was presented to him at a dinner at the Albion tavern on 20 December 1828.
In the section of Porsoniana appended to Alexander Dyce's Table-talk of Samuel Rogers, p. 302, occurs the anecdote that when Richard Porson was introduced to Cogan with the remark that he was intensely devoted to Greek, Porson's reply was, "If Mr. Cogan is passionately fond of Greek, he must be content to dine on bread and cheese for the remainder of his life."
Besides the Fragment on Philosophical Necessity, Cogan wrote: He was the author of several sermons on the deaths of members of his congregation at Cheshunt and Walthamstow, and he read in manuscript and suggested some alterations in Alexander Crombie's Natural Theology (1829).