At this time Trinity College, Dublin, restricted its degrees, fellowships, and scholarships to Anglicans. It was some time after this, in 1873, that all religious requirements were removed from the university. This was not a problem for Salmon, who was an Anglican, but in order to take up the Fellowship that Trinity offered him in 1841 he was required to take holy orders in the Church of Ireland, which indeed he did; being ordained deacon in 1844 and a priest in 1845. In addition to his mathematics position, he was appointed to a divinity lectureship in 1845.
In 1844 Salmon married Frances Anne Salvador, the daughter of the Reverend J L Salvador. They had six children, four boys and two girls, but only two of these six children (the eldest boy and the youngest girl) survived him. In 1848 he was appointed as Donegal lecturer in mathematics at Trinity.
The mathematics department in Trinity College had some outstanding mathematicians on the staff when Salmon joined them in 1841. Hamilton and MacCullagh were on the staff there, as were Charles Graves and Humphrey Lloyd. Although the main topic of interest was synthetic geometry, Salmon only worked in this area for a short time before moving into the area of algebraic geometry.
Salmon became interested in the algebraic approach to geometry taken by Cayley, Sylvester, Hermite and later by Clebsch. He became a close friend of Cayley and Sylvester and discovered, together with Cayley, the 27 lines on the cubic surface. To be more accurate, Cayley discovered these but they were enumerated by Salmon. He also made many discoveries about ruled surfaces and other surfaces, including the idea of the normal singularities of an algebraic surface. As was typical of the time, Salmon's work showed a lack of concern with complete rigour.
A characteristic of Salmon's work was his love of carrying out lengthy calculations. He calculated an invariant of a curve of degree six and published the resulting calculation, which ran to thirteen pages, in the second edition of his treatise on higher algebra which appeared in 1866.
Salmon received many honours for his mathematical contributions. Elected to the Royal Irish Academy in 1843 he received its Cunningham Medal in 1858. He was elected to the Royal Society in 1863 and awarded the Royal Medal of that Society in 1863 and its Copley Medal in 1889. He was also elected to the Institute of France and the academies of Berlin, Göttingen, Copenhagen, and the Accademia dei Lincei of Rome.
From 1848 to 1866, Salmon played a major role in both the departments of mathematics and divinity. We have noted his mathematical contributions over this period above, but we should also note that he had important theological publications. The first of these was his sermon Prayer published in 1849, which was followed by a series of publications of his sermons, for example many are collected in Sermons preached in Trinity College Chapel (1861) and Cathedral and University Sermons (1900). It was generally accepted that his sermons were better read than listened to, since he did not have a strong voice. He also published many articles on issues dividing the Roman Catholic and Anglican churches.
In 1866 Salmon was appointed Regius Professor of Divinity at Trinity College, Dublin. At this time he gave up his mathematics lectureship, and although he continued to work at mathematics, it became more for his own amusement. His last mathematics publication was in 1873 on periods of the recurring decimals of the reciprocals of prime numbers. He published much in the area of theology with works such as The eternity of future punishment (1864), The reign of law (1873), Non-miraculous Christianity (1881), Introduction to the New Testament (1885), The infallibility of the Church (1889), Thoughts on the textual criticism of the New Testament (1897), and a series of articles between 1877 and 1887 on the history of the early Christian Church in the Dictionary of Christian Biography.
In 1871 Salmon was installed as Chancellor of St Patrick's Cathedral and in 1888 he was appointed Provost of Trinity College. He held both positions until his death.
Article by: J J O'Connor and E F Robertson
One of the remarkable geometric results obtained by Salmon: A smooth cubic surface in complex projective 3-dimensional space has exactly 27 complex lines embedded in it. When translated into real space, a smooth cubic surface will have up to 27 real lines embedded in it.
The Theorem of Salmon - "If M, A, B, C are four points on a circle then the circles of diameter [MA], [MB], [MC] intersect in three collinear points."
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