... insomuch that if he continues to prosecute these Studies, as he hath begun, it is not to be doubted but that he will become one of the most skilful Mathematicians that are now living; since at the Age of 22 Years, he hath already attained to so great a Knowledge in those abstruse and difficult Sciences: Wherefore what improvement may we from not expect from the extraordinary Judgement of his riper Years? insomuch that if he continues to prosecute these Studies, as he hath begun, it is not to be doubted but that he will become one of the most skilful Mathematicians that are now living; since at the Age of 22 Years, he hath already attained to so great a Knowledge in those abstruse and difficult Sciences: Wherefore what improvement may we from not expect from the extraordinary Judgement of his riper Years?For the review that this statement is taken from, see THIS LINK.
Rather remarkably Raphson was made a member of the Royal Society in 1689 when, if our date of birth is correct, he was only 21 years of age. He was proposed for election by Edmond Halley at the meeting of the Royal Society on Wednesday 27 November 1689. Halley had himself been elected to the Royal Society at the early age of 22. Raphson was elected a fellow at the meeting on Saturday 30 November 1689 and, four days later, he signed the Charter Book of the Society. At the same time, he signed a bond to pay his quarterly subscription to the Royal Society on which he described himself as:-
Joseph Raphson of London Gent.Raphson's election to that Society was on the strength of work which was published in his book Analysis aequationum universalis
Mr Halley related that Mr Raphson had Invented a method of Solving all sorts of Equations, and giving their Roots in Infinite Series, which Converge apace, and that he had desired of him an Equation of the fifth power to be proposed to him, to which he returned Answers true to Seven Figures in much less time than it could have been effected by the Known methods of Vieta.This method invented by Raphson and described in his book Analysis aequationum universalis
Mr Raphson's Book was this day produced by E Halley, wherein he gives a Notable Improvement of the method of Resolution of all sorts of Equations Showing how to Extract their Roots by a General Rule, which doubles the known figures of the Root known by each Operation, So that by repeating 3 or 4 times he finds them true to Numbers of 8 or 10 places. The Society being highly pleased with this his performance Ordered him their thanks with their Desires, that he would please to Continue to prosecute those Studies, wherein he hath been so Successful.A copy of Raphson's book was presented as "a gift from the author" to the Society on Saturday 17 January 1691.
You can see some information about the Newton-Raphson method at THIS LINK.
Raphson's relation to Newton is important but not particularly well understood. In  Copenhaver writes:-
Raphson was one of the few people whom Newton allowed to see his mathematical papers. As early as 1691, he and Edmond Halley were involved in plans to publish Newton's work of the early 1670's on quadrature of curves, a project fulfilled only in 1704, and then in a much different form. In 1711, Roger Cotes and William Jones arranged for Raphson to see some of Newton's papers "... pertinent to his design of writing an History of the Method of Fluxions."We note that the authors of  say they have seen no evidence to justify the statements by Copenhaver in the first two sentences of the above quote. Raphson did indeed write his History of Fluxions which did not appear until 1715, after Raphson had died, under the title Historia fluxionum. It is unclear how pleased Newton was with this work despite its clear position in favour of Newton's claims over those of Leibniz. Certain letters which had passed between Newton and Leibniz appeared as an appendix to a reprint of Raphson's book in 1716-1718. (This is discussed further at the end of this biography.) Immediately a row broke out and Johann Bernoulli showed his anger. An attempt was made by Newton to calm things down when he wrote to Johann Bernoulli saying:-
I stopt [Raphson's 'History of Fluxions'] coming abroad for three or four years.However, Newton admitted in a letter to Pierre Varignon that he was responsible for the letter being added to Raphson's book:-
When I heard that Mr Leibniz was dead I caused what had passed between him and me to be printed at the end of Raphson's book because copies thereof had been dispersed by Mr Leibniz.This was not Raphson's only publication relating to Newton's work. He translated Newton's algebraic work from Latin to English. Newton's Arithmetica universalis
Early in his career Raphson published a mathematical dictionary. In 1691, the year Raphson was elected to the Royal Society, Jacques Ozanam published Dictionnaire mathématique. Raphson produced his shorter version A mathematical dictionary in 1702 which is:-
A mathematical dictionary or a compendious explication of all mathematical terms, abridg'd from Monsieur Ozanam and others ... written by J Raphson FRS.Raphson published a second edition of his analysis book and, at the same time, appended his article De spatio reali
De spatio reali
Raphson's ideas of space and philosophy were based on Cabalist ideas. The Cabala was a Jewish mysticism which was influential from the 12th century on. It was an oral tradition and initiation into its doctrines and practices was passed on. Cabala developed several basic doctrines which were strong influences on Raphson's philosophical thinking. The doctrines included the withdrawal of the divine light, thereby creating primordial space, the sinking of luminous particles into matter and a "cosmic restoration" that is achieved by Jews through living a mystical life.
In these two works by Raphson De spatio reali
Let us now say a little about the date of 1712 that we have given for Raphson's death. Cotes corresponded with Raphson in 1709 and, writing to William Jones in February 1711, he mentions Raphson whom he certainly believes to be alive at that time. When Raphson's Historia fluxionum
The history of fluxions, showing in a compendious manner the first rise of, and various improvements made in that incomparable method. By (the late) Mr Joseph Raphson,This shows clearly that by 1715 Raphson had died. There is further evidence for the 1712 date but it is a little confusing. We mentioned above that after Leibniz died in November 1716, Newton produced an appendix to Raphson's Historia fluxionum
The difficulty of finding details of Raphson's life are puzzling as the authors of  write:-
It is quite curious that other biographical details about Raphson are so hard to find. He was very well educated, and yet we have been unable to trace his school, but we do know that it was not Westminster ... His middle books are rather heavy-going theologically oriented affairs which display an extraordinary degree of scholarship (including a knowledge of the cabala) normally to be found only in a man with a university education, yet there remains no evidence that he actually studied at either Oxford or Cambridge, and no reason to think that he went elsewhere either. His apparent wealth would also suggest that we should be able to find poll records or a will, but so far these have not been traced. ... the possibility that he ... was of Irish extraction seems quite real, and could go a long way in explaining why he left so few traces in England.
Article by: J J O'Connor and E F Robertson