As well as his life as Church of England clergyman, Craig also had a life as a mathematician. He tutored mathematics taking pupils at his home, became a friend of Newton, continued his contacts with David Gregory, was friendly with Halley and de Moivre, and corresponded with other Scottish mathematicians such Maclaurin. Craig published three major mathematical works which contain the earliest example of the dy/dx notation of Leibniz in Britain and also contained the integration symbol ∫ . While he was still a student in Edinburgh, Craig published Methodus figurarum lineis rectis et curvis comprehensarum quadraturas determinandi Ⓣ which contains Leibniz's dy/dx notation. This notation is also used in the work he published in 1693, Tractatus mathematicus de figurarum curvilinearum quadraturis et locis geometricis Ⓣ which was the first text published in England to contain the integration symbol ∫ . Dale writes :-
The standard of his work was such that he was noted as a mathematician of the first order ... and the "Acta Eruditorum" of Leipzig ranked him among the originators of the calculus (after Leibniz, but before Newton).Craig published several more papers on topics such as the logarithmic curve, the curve of quickest descent, and on the quadrature of figures. He published eight papers on these topics between 1697 and 1710 in the Philosophical Transactions of the Royal Society and was elected a fellow of the Royal Society on 30 November 1711. However, he was involved in a dispute with Jacob Bernoulli over the calculus and he also had a dispute with Tschirnhaus.
In 1699 he published Theologiae Christianae Principia Mathematica Ⓣ which applies probability to show that the evidence of the truth of the gospels is diminished through time. He claimed that it reaches 0 in the year 3144, and so "proves" that this is an upper bound for the second coming. Stigler examines this work in detail in  claiming that this is an "underappreciated book" containing :-
... a formula tantamount to a logistic model for posterior odds: that is, Craig's probability should be understood as the logarithm of the ratio of the probability of the historical testimony as received at the present time, given the historical hypothesis in question, to the probability of the same testimony, given the negation of that hypothesis.These arguments appear in the first two chapters of the Theologiae Christianae Principia Mathematica Ⓣ while the remaining four chapters give a mathematical argument in support of Pascal's wager, namely that:-
... if God does not exist, the sceptic loses nothing by believing in him; but if he does exist, the sceptic gains eternal life by believing in him.As an aside, I [EFR] must say I have never found Pascal's argument (and that of Craig) convincing since for all we know God may grant the honest sceptic eternal life but find it totally unacceptable that someone chooses to believe in Him purely through self-interest.
In 1718 Craig published a work on optics Quibus subjunguntur libri duo de optica analytica Ⓣ. It is worth noting that despite using Leibniz's notation for the calculus in his earlier works, in this later one Craig used Newton's fluxion notation.
We left our description of Craig's church career at the point where he was vicar of Potterne in Wiltshire. In 1696 he became vicar of Gillingham Major while on 2 November 1708 he was also given a stipend from the revenue of Durnford and became a canon at Salisbury Cathedral. His brother William, who had been held the spiritual charge of Gillingham, died in 1721 and on 28 June 1726 Craig was appointed to the Gillingham role formerly held by William. In the last part of his life Craig went to London in the hope that his mathematical abilities would be noticed. He was buried on 14 October in the churchyard of St James's, Clerkenwell, London.
Article by: J J O'Connor and E F Robertson