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In 1871 Muir became an assistant at Glasgow University, then from 1874 to 1892 he was mathematics and science master at Glasgow High School. In 1882 he published Treatise on the theory of determinants while in 1890 he published History of determinants which was to become the first part of a major five volume life's work. Also in 1890 Muir was elected to a Fellow of the Royal Society, then two years later he went to the Cape, South Africa, as Superintendent General of Education. There he was to introduce many educational reforms.
Muir was knighted in 1910, four years after he had reworked his 1890 publication to become Volume 1 of History of determinants which covered the origins to Leibniz in 1840. The remaining volumes were Volume 2 1840-1860 (1911), Volume 3 1860-1880 (1920), Volume 4 1880-1900 (1923), Volume 5 1900-1920 (1929). Muir was working on Volume 6 1920-1940 at the time of his death was was somewhat optimistic since he would have been over 100 years of age before it could have been completed.
In 1931, at the age of 87, he showed surprising abilities to keep with the modern flavour of his subject when he wrote that he
welcomed the light matrix proofs in contrast to the heavy footed methods of thirty-five years ago.
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