Simpson had lodgings in Nuneaton with a lady by the name of Swinfield whom he married in 1730. They had one daughter Elizabeth born in 1736 and one son Thomas born in 1738. In fact Simpson and his wife had left Nuneaton before his children were born. The reason which has been reported by his biographers is as follows. He :-
... had to flee to Derby in 1733 after he or his assistant had frightened a girl by dressing up as a devil during an astrology session.This would certainly fit in with the fact that he was known as the:-
... oracle of Nuneaton, Bosworth and the environs.Exactly how long he remained in Derby is unknown but we do know that from 1736 he was living in London with his family. He was an early member of the Spitalfields Mathematical Society being one of 49 members in 1736. This Society operated as a working men's club and we know that it was a natural choice for a weaver who taught mathematics since of the members by 1744:-
... about half were weavers, and the rest were typically brewers, braziers, bakers, bricklayers.Simpson was the most distinguished of a group of itinerant lecturers who taught in the London coffee houses. This may seems strange but in fact at this time coffee houses were sometimes called the Penny Universities because of the cheap education they provided. They would charge an entrance fee of one penny and then while customers drank coffee they could listen to lectures. Different coffee houses catered to specific interests such as art, business, law and mathematics. For example De Moivre used Slaughter's Coffee House in St Martin's Lane as a base during these years, and William Jones, who was a friend of Simpson, was able to make a living lecturing in coffee houses such as Child's Coffee House in St Paul's Churchyard.
In 1743 Simpson was appointed as the head of mathematics at the Royal Military Academy at Woolwich. In fact this Academy was founded only two years before Simpson took up the post and his appointment there had an impact on the mathematical topics he investigated. In particular he began research on engineering problems and problems relating to fortifications. Two years after his appointment, Simpson was elected a fellow of the Royal Society. While we are describing the honours which he received, we should note that he was also elected a fellow of the Royal Swedish Academy of Sciences in 1758.
From 1737 Simpson began to write texts on mathematics, publishing A New Treatise of Fluxions in that year :-
This was a high-quality textbook devoted to the calculus of fluxions, the Newtonian version of the infinitesimal calculus. The topic was advanced -- it was no trivial exercise to write such a book in the 1730s, when the calculus was mastered by only a few mathematicians in Europe.Simpson is best remembered for his work on interpolation and numerical methods of integration. However the numerical method known today as "Simpson's rule", although it did appear in his work, was something he learned from Newton as Simpson himself acknowledged. By way of compensation, however, the Newton-Raphson method for solving the equation f (x) = 0 is, in its present form, due to Simpson. Newton described an algebraic process for solving polynomial equations which Raphson later improved. The method of approximating the roots did not use the differential calculus. The modern iterative form xn+1 = xn - f (xn) / f '(xn) is due to Simpson, who published it in 1740.
He also worked on probability theory and in 1740 published The Nature and Laws of Chance. Much of Simpson's work in this area was based on earlier work of De Moivre. In fact he was involved in a dispute with De Moivre over issues of priority on the topic of probability and annuities. He worked on the Theory of Errors and aimed to prove that the arithmetic mean was better than a single observation. His justification of this appeared in his 1757 memoir An attempt to show the advantage arising by taking the mean of a number of observations in practical astronomy.
Simpson published Mathematical Dissertations in 1743 which discussed the attraction of the solid obtained by rotating an ellipse around one of its axes. His two volume work The Doctrine and Application of Fluxions in 1750 contains work of Cotes and is considered by many to be the best work on Newton's version of the calculus published in the 18th century. Problems in astronomy such as the precession of the equinoxes were discussed by Simpson in Miscellaneous Tracts (1757).
In 1754 he became editor of the Ladies Diary. He had published in the Ladies Diary from the time he came to London in 1736. He answered problems posed in this publication, but used a variety of pseudonyms such as Marmaduke Hodgson, Hurlothrumbo, Kubernetes, Patrick O'Cavannah, and Anthony Shallow. It was his obvious mathematical skills demonstrated in these solutions which first brought his to the attention of other mathematicians of the day. Other periodicals which he published in were the Gentleman's Magazine, Miscellanea Curiosa Mathematica, and the Gentleman's Diary.
In  Stigler describes an event which occurred near the end of Simpson's life:-
A newly discovered manuscript fragment shows that Thomas Simpson and Roger Boscovich met in 1760, and that Boscovich posed a problem in least deviations regression to Simpson. ... Simpson's attempt at an analytical solution is interpreted.The following description of Simpson by Charles Hutton (made 35 years after Simpson's death) is interesting :-
It has been said that Mr Simpson frequented low company, with whom he used to guzzle porter and gin: but it must be observed that the misconduct of his family put it out of his power to keep the company of gentlemen, as well as to procure better liquor.It would be fair to note that others described Simpson's conduct as irreproachable.
Article by: J J O'Connor and E F Robertson