His career as a professional astronomer began in 1870, when he was elected Savilian professor of astronomy at Oxford.
He was elected fellow of Balliol in 1850 and Savilian professor of geometry in 1861, and in 1874 was appointed keeper of the university museum.
In 1657 he became professor of astronomy at Gresham College, and in 1660 was elected Savilian professor of astronomy at Oxford.
Sir Henry Savile (1549-1622) thereupon appointed him in 1619 to the Savilian chair of astronomy just founded by him at Oxford; Bainbridge was incorporated of Merton College and became, in 1631 and 1635 respectively, junior and senior reader of Linacre's lectures.
In 1643 he was appointed to the Savilian professorship of astronomy at Oxford, but he was deprived of his Gresham professorship for having neglected its duties.
In 1648 he lost both his fellowship and his Savilian chair on account of his adherence to the royalist party.
Notwithstanding this act of opposition, he was in June 1649 appointed Savilian professor of geometry at Oxford.
The titles in the order adopted, but with date of publication, are as follows: "Oratio inauguralis," on his appointment (1649) as Savilian professor (1657); "Mathesis universalis, seu opus arithmeticum philologice et mathematice traditurn, arithmeticam numerosam et speciosam aliaque continens" (1657); "Adversus Meibomium, de proportionibus dialogus" (1657); "De sectionibus conicis nova methodo expositis" (1655); "Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadraturam aliaque difficiliora matheseos problemata" (1655); "Eclipsis solaris observatio Oxonii habita 2° Aug.
Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, welcomed the Descriptio with enthusiasm.
In 1654 Seth Ward (1617-1689), the Savilian professor of astronomy, replying in his Vindiciae academiarum to some other assaults (especially against John Webster's Examen of Academies) on the academic system, retorted upon Hobbes that, so far from the universities being now what he had known them in his youth, he would find his geometrical pieces, when they appeared, better understood there than he should like.
Ward's colleague, the more famous John Wallis, Savilian professor of geometry from 1649, had been privy to the challenge thrown out in 1654, and it was arranged that they should critically dispose of the De corpore between them.
He, however, resigned his ecclesiastical preferments in 1721, on his appointment to the Savilian professorship of astronomy at Oxford, while as reader on experimental philosophy (1729-1760) he delivered 79 courses of lectures in the Ashmolean museum.
On his return from a journey to Dalmatia, for the purpose of selecting and fortifying the port of Trieste, he was nominated, November 1703, Savilian professor of geometry at Oxford, and received an honorary degree of doctor of laws in 1710.
In 1883 he was chosen to succeed Henry Smith in the Savilian chair of geometry at Oxford, and there he produced his theory of reciprocants, largely by the aid of his "method of infinitesimal variation."