I am indebted to Clare Hopkins, Archivist of Trinity College, Oxford, for her research efforts culminating in the below information (including the translation from Latin) regarding Rev. Chapman.
The above is Dr. Chapman's original entry, in his own hand, in Trinity
College's Admission Register, which translates as
I, Joseph Chapman, son of Joseph Chapman of Daglingworth, Gloucestershire, born at Cirencester in the same county, was admitted a commoner of the third order, under the tuition of Master Parker, 10 October 1759.
Chapman was elected to a scholarship (his fees and expenses all paid
by the college) in June 1760, and to a fellowship in June 1769. These elections
always took place on Trinity Monday, which is a moveable feast in the
religions calendar, but always in June. He was then elected to the college Presidency at the young age of 33, in 1776, and he retained the Headship of the college until his death in 1808.
There is a brief reference to his presidency in the History of Trinity
College, by H E D Blakiston, which was published in 1898. He was obviously
well-liked but not particulary active, although it seems he was embroiled
in one or two controversies with some of the junior fellows during his presidency. He is described as 'a tall, dignified, well wigged Head of House who preferred a quiet easy life'.
There is no record in the Trinity archive of Chapman's marraige, or of his children. He died in 1808 and was buried in antechapel of the college chapel.
Blakiston also says that Chapman was Rector of Daglingworth 1775-97 (many Oxford dons held this kind of appointment, as the actual work in the parish would have been done by a curate). It is also noted that he succeeded his father in this rectorship, and was in turn succeeded by his brother John.