On Ridgley's death (27 March 1734) he succeeded him as theological tutor, handing over his previous duties to Joseph Densham, one of his pupils.
His reputation as a tutor, especially in natural science, was great; it appears that Thomas Secker attended his classes (in 1716–17, at the time when he was turning his thoughts towards medicine as a profession).
He enjoyed the friendship of Sir Isaac Newton, through whose influence he was elected Fellow of the Royal Society, whose Philosophical Transactions he was employed in abridging.
Of his theological work nothing remains; on 13 February 1735 he took part with Samuel Chandler and Jeremiah Hunt, in an arranged debate with two Roman Catholic priests, at the Bell Tavern in Nicholas Lane.
Eames, who was unmarried, died suddenly on 29 June 1744, a few hours after giving his usual lecture.