On the 1st of October 1824 he was elected fellow of Trinity, and in December 1826 was appointed Lucasian professor of mathematics in succession to Thomas Turton.
Johnson, whose chief asset was the MS. tragedy of Irene, was at first the host of his former pupil, who, however, before the end of the year took up his residence at Rochester with John Colson (afterwards Lucasian professor at Cambridge).
In 1701 he resigned his living to become deputy at Cambridge to Sir Isaac Newton, `Thom two years later he succeeded as Lucasian professor of mathematics.
From 1828 to 1839 Babbage was Lucasian professor of mathematics at Cambridge.
He wrote a paper Analysis per Equationes Numero Terminorum Infinitas, which he put, probably in June 1669, into the hands of Isaac Barrow (then Lucasian professor of mathematics), at the same time giving him permission to communicate the contents to their common friend John Collins (1624-1683), a mathematician of no mean order.