In 1860, Tait succeeded his old master, James D. Forbes, as professor of natural philosophy at the University of Edinburgh, and occupied the Chair until shortly before his death. The first scientific paper under Tait's name only was published in 1860. His earliest work dealt mainly with mathematical subjects, and especially with quaternions, of which he was the leading exponent after their originator, William Rowan Hamilton. He was the author of two text-books on them—one an "Elementary Treatise on Quaternions" (1867), written with the advice of Hamilton, though not published till after his death, and the other an "Introduction to Quaternions" (1873), in which he was aided by Philip Kelland (1808–1879), one of his teachers at the University of Edinburgh. Quaternions was also one of the themes of his address as president of the mathematical section of the British Association for the Advancement of Science in 1871.