William Neile was born at Bishopsthorpe in the house of his grandfather who was Archbishop of York. William's father was Sir Paul Neile (1613-1686) and his mother was Elizabeth Clarke, the daughter of Gabriel Clarke, the Archdeacon of Durham. Paul Neile, one of the twelve founders of the Royal Society, is described in  as follows:-
Paul Neile was born at Westminster in 1613 and was admitted as a Fellow Commoner to Pembroke College, Cambridge, in 1627, at the age of 14. He was one of the Ushers of the Privy Chamber to Charles I and was knighted in 1633, when he is described as of Sutton Bonvill, Yorks, N.R. He was M.P. for Ripon in the Short Parliament of 1640. ... We know little of Neile's life during the Commonwealth (1649-60) except that he was living at White Waltham near Maidenhead, evidently a man of means with an interest in astronomy and in the making of telescopes.
Paul Neile recorded the births of five of his children, all born at Bishopsthorpe, in the Family Bible (the actual images of the pages of the Bible can be seen at ). The Neiles' first son Richard was born in November 1636 but must have died while a baby since a third son born in June 1640 was also named Richard. William, who was therefore his parents' second son, is recorded as born 3/4 of an hour after 11 o'clock at night on 16 December 1637. This is at odds with ,  and  which all give his date of birth as 7 December 1637 and state that he was his parents' eldest son. Paul Neile also recorded in the Bible that a daughter Elizabeth was born in March 1639 and a second daughter Mary in October 1641.
William Neile entered Wadham College, Oxford, in 1652 (but did not matriculate until 1655) where he was taught mathematics by John Wilkins and Seth Ward. He was a gentleman-commoner, meaning that he paid the highest fees and was ranked near the top of the social order just below the nobles. Gentleman-commoners had many privileges enjoying fine suites of rooms in College, and sat with the College Fellows at meals and in the common rooms. Certainly Neile was fortunate in being part of a family that was in the forefront of scientific work for certainly while Neile was a student, his father was observing with Christopher Wren in the observatory he had constructed on the roof of his house, the 'Hill House', at White Waltham. Paul Neile was also building a telescope for Gresham College at this time. In 1657 William Neile became a pupil of law at the Middle Temple in London. He went on to become a member of the privy council of King Charles II.
In 1657, while still a student at Oxford, he became the first person to find the arc length of an algebraic curve when he rectified the semicubical parabola. He communicated his results to William Brouncker and Christopher Wren at the Gresham College Society, the Society based at Gresham College, London, which a few years later became the Royal Society. Neile's work on this appeared in John Wallis's De Cycloide in 1659. Stedall gives further details in :-
In 1657, William Neile, a young student at Wadham College, Oxford, found the rectification of the semicubical parabola (in modern notation 9y2= 4kx3)by a method that was geometric but involved a comparison of sums of infinitely small quantities. Wallis was easily able to make Neile's proof algebraic using the notation defined in 'De sectionibus conicis', while William Brouncker went further and came up with a formula for the length of a portion of the curve in terms of its coordinates. At about the same time, Hendrick van Heuraet in the Netherlands arrived independently at a general method of rectification, and applied it to the semicubical parabola, and in 1659 Fermat rectified both the semicubical parabola and the cycloid. Wallis later claimed that all these attempts were based on the hints he had given in the 'Arithmetica infinitorum':
"And I do not at all doubt that this notion there hinted, gave occasion (not to Mr Neile only, but) to all those others (mediately and immediately,) who have since attempted such Rectification of Curves (nothing in that way having been attempted before)."
It is true that Wallis had outlined a method of rectification in the 'Arithmetica infinitorum', and Neile may have been inspired by it, but Neile's method was expressed in traditional geometric terms and he handled a curve that Wallis had not thought about at all.
Neile was elected a Fellow of the Royal Society on 7 January 1663, one of the first members of this Society. On 11 April 1666 he became a member of the Council of the Royal Society joining his father who had been a Council member since 1661. As a Council member, William Neile proposed changes to the Society, in particular the setting up of a library:-
... let there be every year a sum of money laid aside out of the Society's stock to be laid out at the discretion of the Council in books whose chief scope is Natural Philosophy or Mathematics for the use of the Society.
He sent Hypothesis of Motion, a work on the theory of motion, to the Society on 29 April 1669 after taking part in many debates in the Society on that topic. Interesting information about Neile can be found in Henry Oldenberg's correspondence  and Neile's paper (published by the Society in May 1669) is contained in Volume 5 of that work. Oldenberg was the secretary of the Royal Society and his correspondence contains several letters from Neile. For example, in relation to the debates on motion we have just mentioned, Neile wrote:-
I wish Dr Wren would explain his principles a little more fully but he is against finding a reason for the experiments of motion (for ought I see) and says that the appearances carry reason enough in themselves as being the law of nature. I think it is the Law of nature that they should appear but not without some causes, but all those things perchance will be better determined hereafter.
In Hypothesis of Motion, Neile defined as one body all matter sharing the same motion and studied impacts assuming that the colliding bodies were cubes and that :-
... the whole square surface of the one meets in the same instant of time with the whole square surface of the other.
In January 1668 Neile had written a letter to Oldenburg complaining that the country was putting far too much of its resources into military preparations rather than supporting science. He wrote :-
I am sorry we should want any assistance to philosophy, since the Great Turk or the like kind of Hectors can have it may be 100,000 men to besiege a town or make a conquest perchance of less importance by much than to discover the use of the air or of the commonest thing in nature.
As well as his mathematical work Neile made astronomical observations using instruments on the roof of his father's house, the 'Hill House' at White Waltham in Berkshire. He died in this house at the age of 32 and was buried in the local parish church. Thomas Hearne, the English historian and antiquarian who was himself born in White Waltham 8 years after Neile's death, describes him as follows:-
He was a virtuous, sober pious man, and had such a powerful genius to mathematical leaning that had he not been cut off in the prime of his years, in all probability he would have equalled, if not excelled, the celebrated men of that profession. Deep melancholy hastened his end, through his love for a maid of honour, to marry whom he could not obtain his father's consent.
Some time before his death he suffered an attack of jaundice and this may have been related to the cause of his death.
Article by: J J O'Connor and E F Robertson