1845 he was appointed divinity lecturer as well, and his long life was devoted to these two diverse lines of study. For many years he was a college tutor ; from 1848 to 1866, the period during which his mathematical books were written, he was Donegal lecturer in mathematics. Salmon's first mathematical paper, 'On the properties of surfaces of the second degree which correspond to the theorems of Pascal and Brianchon on Conic Sections,' was pubhshed in the 'Philosophical Magazine' in 1844. In 1847 there appeared his 'Conic Sections,' the work which made him known as a mathematician to a wide circle (6th edit. 1879). Admirably arranged, and constructed with an unerring sense of the distinction between important principles and mere details, it exhibited more fully than any other book of the time at once the power of the Cartesian co- ordinates and the beauty of geometrical method ; and for half a century it was the leading text-book on its subject. It was followed in 1852 by a treatise on the 'Higher Plane Curves' (3rd edit. 1879), a subject of which little was then known, and which was introduced to the ordinary student by Salmon's labours. The investigations of Cayley and Sylvester into the invariants of quantics were beginning to attract attention ; and Salmon proceeded to apply their results to geometrical theory, the result being his 'Lessons Introductory to the Modern Higher Algebra' (1859; 4th edit. 1885), in which he incorporated much original matter. Finally in 1862 appeared the 'Geometry of Three Dimensions' (5th edit. 2 vols. 1912), in which the sections upon the general theory of surfaces are specially remarkable (the work was translated into French, German, and Spanish). Upon these four treatises his fame as a mathematician rests, while many minor papers by him appeared in the learned journals. Salmon's
methods made little use of the calculus, or of the quaternion analysis invented by his contemporary. Sir W. R. Hamilton [q.v.]; nor, again, did he ever handle the non-Euchdean geometry. His strength lay in his complete mastery of geometric and algebraic processes, and this, coupled with his indefatigable industry as a calculator, enabled him to produce original work of permanent value. In later life, the theory of numbers fascinated him ; and he spent many odd half-hours in determining the number of figures in the recurring periods in the reciprocals of prime numbers. His last mathematical paper was upon this subject (' Messenger of Mathematics,' 1873), but he never pubHshed his latest results, and he used to speak of his calculations as a useless amusement.
Salmon's mathematical labours by no means exhausted his energies, and he took a large share in the work of the Divinity School of Trinity College from 1845 to 1888. He proceeded B.D. and D.D. in 1859, and from 1866 to 1888 he was regius professor of divinity. He played an active part in the reconstruction of the Irish Church after its disestablishment in 1870, and enjoyed a unique position in the General Synod and as a member of the Representative Church body, his skill as a debater and his ability in the management of the church's finance being equally remarkable.
Salmon's first publication on a theological subject was a sermon on Prayer (1849), the precursor of a long series of printed discourses. His preaching always commanded attention, but his sermons (of which five volumes were published) were better to read than to hear, for his voice was hardly effective in a large building. In 1 852 Archbishop Whately made him an examining chaplain, and the archbishop's influence upon Salmon's theological opinions seems to have been considerable. Both men were strong Protestants, and viewed the rise of the Oxford movement with suspicion and dislike, Salmon co-operating with Whately and others in the issue of 'Cautions for the Times' (1853), intended as a counterblast to the famous ' Tracts.' He was also a frequent contributor to the 'Catholic Layman,' which dealt with the Roman catholic controversy, and he printed anonymously three short 'Popular Stories' (Dublin 1854) written in the same interest. This preparation bore fruit later on, when, as divinity professor, he lectured on the points at issue between Romanism and Anglicanism ; and his lectures formed the material of 'The Infallibility of the Church' (1889 ; 2nd edit. 1890), a trenchant and brilliant polemic which exhibited his learning, his humour, and the vigour of his controversial methods. Salmon founded no school of theological thought, deeply as he was revered by his pupils, his genius being analytic and even destructive rather than constructive and synthetic ; but his tendency was towards a liberal evangelicalism, which distrusted (and more and more as years went on) the appeal to any authority other than that of the individual conscience.
The studies by which he became most widely known as a divine lay, however, outside the sphere of dogmatic theology,
