Firstly, perhaps it was a career choice made by Salmon, particularly surrounding the vacancy of the Erasmus Smith Professorship of Mathematics at Trinity College in 1862 that demonstrates the beginning of his shift in focus from mathematics to theology. The previous Professor of Mathematics, Charles Graves, resigned this position, and Gow  suggests that although Salmon would have been the worthiest candidate to fill this post, he did not offer himself for the position. He suggests that:-
For whatever reason, Salmon must have already decided to concentrate his energies in theology, instead of mathematics.
Gow  does not speculate on the reason for Salmon's shift in focus to theology. However, he notes that at the same time a vacancy arose in the Divinity department, the Archbishop King's Lectureship in Divinity, with a salary equal to the Mathematical position (£700 per annum). Gow refers to C J Joly's biography of Salmon , who suggests that Salmon was led to believe that William Lee, a more senior theologian in the department, was not going to apply for the position. However, this was not the case, and Lee was chosen over Salmon due to Lee's seniority, as he had been a fellow since 1839. Salmon received his divinity promotion in 1866. However the events in 1862 demonstrate that Salmon's priority was now theological studies and he wished to further his academic career within the Divinity department.
Although I have suggested that the reason for Salmon's shift from mathematics was as a career choice, this still does not provide us a reason as to why this change occurred. Secondly therefore, and perhaps of more significance, Salmon's shift towards theology was likely influenced by the major changes and disputes within the Church of Ireland occurring at this time. This undoubtedly captured Salmon's attention, as he is often described as very outspoken. The discussions reached a climax in The Irish Church Act in 1869 that dissolved the statutory union between the Churches of England and Ireland, and declared that the Church of Ireland ceased to be established by law. This led to major ecclesiastical reform within the Church of Ireland, as reflected in the Preamble and Declaration from the Book of Common Prayer. 1870:-
Whereas it has been determined by the Legislature that on and after the first day of January, 1871, the Church of Ireland shall cease to be established by law; and that the Ecclesiastical Law of Ireland shall cease to exist as Law save as provided in the 'Irish Church Act, 1869;' and it hath thus become necessary that the Church of Ireland should provide for its own regulation.
Thirdly, following on from the previous reason, we should allow for the change of direction taken by Salmon to be from a religious conviction. Salmon came from a strong religious background and was heavily involved in the church, so it should not be forgotten that he was a strongly religious man. Perhaps then it was simply from a personal conviction that this was God's will and leading for him that led Salmon to give priority to his theological studies.
Fourthly, perhaps Salmon had become bored with mathematics, or he felt he had exhausted his ability or had received criticism of his work. Gow notes that some mathematicians found his textbooks to be occasionally "definitely unclear and badly expressed." He completed his four influential textbooks from 1848 to 1862, and during this time he had also produced 40 papers. After this he produced few mathematical publications, with his last paper published in 1873.
It is interesting to note here that Salmon's professor, MacCullagh, committed suicide in 1847, an event that undoubtedly had an impact on Salmon. MacCullagh had written to Babbage in 1842, stating:_
I have grown very stupid of late, and regularly fail at everything I attempt. What the reason may be I cannot tell. But I begin to be of Newton's opinion, that after a certain age, a man may as well give up mathematics.
Without wanting to overstate the impact MacCullagh's suicide may have had upon Salmon, perhaps he was influenced by Newton's sentiments also, and realised that he had exhausted his mathematical ability and decided to give up mathematical studies in favour of his other interests. On the subject of Salmon's shift from mathematics to theological studies, the Dictionary of Scientific Biography records :-
Over the years Salmon became frustrated by the heavy load of tutoring and lecturing, much of it an elementary kind and was disillusioned because he was not made a professor, a promotion that would have relieved him of most of this load and given him more time for research. It must have been this which influenced him, in about 1860, to turn away from mathematics towards theological studies in which he had always been interested Ð and which appeared to offer better prospects of promotion.
Therefore we can assume that a catalogue of factors gradually shifted Salmon's focus from mathematics to theology. This began in 1862 when he failed to secure his mathematics promotion and shortly after this his first major theology work was published in 1864, The Eternity of Future Punishment. He did however still continue to have an interest in mathematics for the next decade or so, as it was not until 1873 that his last mathematical work was published, Periods of the Recurring Decimals of the Reciprocals of Prime Numbers. The pivotal moment of this shift was probably his appointment as the Regius Professor of Divinity at Trinity College in 1866.
In 1871 Salmon was appointed as the Chancellor of St Patrick's Cathedral in Dublin, an important appointment given the critical changes the Church was experiencing at this time. Salmon produced the first of his four major theological works in 1881, on Non-Miraculous Christianity. The second came in 1885, a strong polemic entitled Introduction to the New Testament. From 1877 Ð 1887 he also wrote various articles on early church history for The Dictionary of Christian Biography.
In 1888 the Crown appointed Salmon the Provost of Trinity College. In the next year he produced his third and most controversial and important theological work, The Infallibility of the Church, which was a series of lectures in which he argues against the tradition of Papal Infallibly within the Roman Catholic Church. His fourth important theological work, Thoughts on Textual Criticism of the New Testament, was published in 1897. Salmon served as Provost until his death on 22 January 1904.
I will show below how Salmon's flair in the disciplines of both mathematics and theology complemented each other. Firstly, I will focus on his most famous theological work, his series of lectures on The Infallibility of the Church, to illustrate how his mathematical brilliance was a factor in his theological works, especially in his analytical approach and process orientated approach to find meaning and truth.
On reading Salmon's theological works I have found that they are written in a highly analytical style. He often sets his arguments out in what could be described as mathematical proofs, for example, setting out statements and proving or disproving them, or showing how two statements on a subject will contradict the other therefore disproving the overall concept. Some of his lectures have mathematical titles, for example, Milner's Axioms and The Argument in a Circle Ð here Salmon addresses the concept of proof and examines the methods that have been used to 'prove' certain doctrines. He concludes in G Salmon, The Infallibility of the Church, 1888 edition, Lecture 4, that in some cases:-
It is not possible ... without being guilty of the logical fallacy of arguing in a circle.
Salmon largely structures his theological writing around the absolutes of true and false: this is one of the reasons why his writing is so controversial. He often confidently demonstrates through argumentation how important concepts of the Roman Catholic Church are inherently incorrect. He usually sets out his arguments in the form of numbered lists, and frequently appeals to 'logic.' He is extremely thorough in his examination of all sides of an issue and offering a well-supported conclusion.
The following are some examples of this style, which can be traced in through all his theological writing. In G Salmon, The Infallibility of the Church, 1888 edition, Lecture 2 he writes:-
Thus, as long as anyone really believes in the infallibility of his Church, he is proof against any argument you can ply him with. Conversely, when faith in this principle is shaken, belief in some other Roman Catholic doctrine is sure also to be disturbed; for there are some of these doctrines in respect of which nothing but a very strong belief that the Roman Church cannot decide wrongly will prevent a candid inquirer from coming to the conclusion that she has decided wrongly. This simplification, then, of the controversy realises for us the wish of the Roman tyrant that all his enemies had but one neck. If we can but strike one blow, the whole battle is won.
An example of an argument won by providing a counter example and thus disproving a greater theory appears in G Salmon, The Infallibility of the Church, 1888 edition, Lecture 5:-
From the title-page, as it appears on the paper cover of each, the two books appear to be both of the twenty-first thousand; but when we open the books, we find them further agreeing in the singular feature, that there is another title-page which describes each as of the twenty-fourth thousand. But at page 112 the question and answer which I have quoted are to be found in the one book, and are absent from the other. It is, therefore, impossible now to maintain that the faith of the Church of Rome never changes, when it is notorious that there is something which is now part of her faith which those who had a good right to know declared was no part of her faith twenty years ago.
George Salmon's life and work is an excellent example of how complementary the study of both mathematics and theology can be, as I have demonstrated above. There has been much written on this matter, and there is little scope in this conclusion to expand on this topic properly. However, I would like to conclude by mentioning some of the commonalities between the two disciplines of mathematics and theology. Pythagoras said that:-
Number rules the universe ... Number is the within of all things.
Geometry is knowledge of the eternally existent.
It is interesting that often, pure mathematics is misunderstood for its relevance to everyday life, yet the same statement is often made of matters of faith. The method of proof by induction can often be detected as the structure behind many theological arguments. A good example is when theologians attempt to outline an argument to show that God exists. The cosmological method for this begins with the statement that there is a created world. Let us call this statement A1. A1 is taken as fact. Then the statement A2 is made, something like the world is made up of contingent entities; using statement A1 this is also taken to be true; the world had a starting point, its creation, through whatever process. So A2 is true. The argument continues to the conclusion that this creative process must have begun with a non-contingent being, and the inductive step is taken to assume that this is the 'being that which we call God.' Similarly, the ontological argument for the existence of God uses the principle of the necessity of an uncaused cause, The created world is like a chain of caused evolutionary events, and this process rather than being an infinite regression series had a starting point, an uncaused cause, again 'the being that which we call God.'
Another example is that of infinity. The concept of God as infinite is a foundational principle of Christian belief. What is meant by infinite is infinity in both directions; that there is no beginning and no end to God, respective to many of the properties essential to God's character; infinite existence or timelessness, power, knowledge, goodness, love etc.