**Joseph Raphson**'s parents were Ruth and James Raphson, although this cannot be confirmed with absolute certainty. James Raphson's will is dated October, 1689, and he was buried in Pinner churchyard in Middlesex. Joseph's life can only be deduced from a number of pointers. No obituary of Raphson seems to have been written and we can now only piece together details about his life from records which exist such at University of Cambridge records and records of the Royal Society. It is through the University of Cambridge records that we know that Raphson graduated with an M.A. from Jesus College Cambridge in 1692 but it is almost certain that he never studied at Cambridge since the M.A. is by Royal warrant. His birth date was given as 1649 by Florian Cajori, but a date of 1668 seems much more likely. In fact someone of the name "Joseph Raphson" was baptised in the Parish Church in Pinner, Middlesex in the 1660s and, again it cannot be proved with certainly that this is the Joseph Raphson of this biography but it does seem highly probable. Our date of 1668 seems to be essentially confirmed by the following statement made in 1891:-

For the review that this statement is taken from, see THIS LINK.... 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 of22Years, 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 of22Years, 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?

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:-

Raphson's election to that Society was on the strength of work which was published in his bookJoseph Raphson of London Gent.

*Analysis aequationum universalis*Ⓣ in 1690. Edmond Halley had reported on Raphson's work at the meeting of the Society on 30 July 1690 as recorded in the Minutes:-

This method invented by Raphson and described in his bookMr 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.

*Analysis aequationum universalis*Ⓣ is now called the Newton method (or the Newton-Raphson method) for approximating the roots of an equation. In

*Method of Fluxions*Newton describes the same method and, as an example, finds the root of

*x*

^{3}- 2

*x*- 5 = 0 lying between 2 and 3. Although written in 1671 it was not published until 1736, so Raphson published the result nearly 50 years before Newton. The Minutes of the meeting of the Royal Society on Wednesday 17 December 1690 contain a reference to Raphson's

*Analysis aequationum universalis*Ⓣ:-

A copy of Raphson's book was presented as "a gift from the author" to the Society on Saturday 17 January 1691.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 repeating3or4times he finds them true to Numbers of8or10places. 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.

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 [2] Copenhaver writes:-

We note that the authors of [3] 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 titleRaphson was one of the few people whom Newton allowed to see his mathematical papers. As early as1691, he and Edmond Halley were involved in plans to publish Newton's work of the early1670's on quadrature of curves, a project fulfilled only in1704, and then in a much different form. In1711, 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."

*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:-

However, Newton admitted in a letter to Pierre Varignon that he was responsible for the letter being added to Raphson's book:-I stopt[Raphson's 'History of Fluxions']coming abroad for three or four years.

This was not Raphson's only publication relating to Newton's work. He translated Newton's algebraic work from Latin to English. Newton'sWhen 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.

*Arithmetica universalis*Ⓣ was translated by Raphson and was published as

*Universal arithmetick*in 1720 after Raphson's death.

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:-

Raphson published a second edition of his analysis book and, at the same time, appended his articleA mathematical dictionary or a compendious explication of all mathematical terms, abridg'd from Monsieur Ozanam and others ... written by J Raphson FRS.

*De spatio reali*Ⓣ which is an application of mathematical reasoning to theological issues. Raphson wrote a second theological work

*Demonstratio de deo*Ⓣ in 1710. You can read detailed reviews of both of these works at THIS LINK.

*De spatio reali* Ⓣ discusses space and in it Raphson talks of 'real space' which he thinks of as being independent of the mind that perceives it. He discusses the infinite, distinguishing between the potentially infinite and the actual infinite. In discussing motion he argues that space is infinite but the collection of moving objects in it is finite.

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* Ⓣ and *Demonstratio de deo* Ⓣ, cosmology, natural philosophy, mathematics and his Cabalist beliefs combine. Of course his religious beliefs greatly influenced all his thinking. Newton's views of space were strongly influenced by Christian beliefs, and possible just slightly by his interaction with Raphson.

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* Ⓣ was published in 1715 the title page states:-

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'sThe 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,

*Historia fluxionum*Ⓣ but, relevant to the date of Raphson's death, retained the same 1715 publication date on the work. Newton's appendix states clearly that Raphson had died three years earlier. If we assume that Newton deliberately kept the 1715 date on the book despite the appendix being added one or even two years later, then one assumes that he would write that Raphson died three years earlier if he died in 1712. A date of late 1712 would also be consistent with the fact that Raphson's name appears in the 1712 list of fellows of the Royal Society but not in the 1713 list.

The difficulty of finding details of Raphson's life are puzzling as the authors of [3] 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*